# The compact symplectic group $Sp(n)$

**The compact symplectic group** $Sp(n)$ is the group of length-preserving linear transformations of $n$-dimensional quaternionic space. In one line,

$$Sp(n)=\{A\in M_n(\mathbb H):A^*A=I\}.$$

This looks like the familiar definition $O(n)=\{A\in M_n(\mathbb R):A^TA=I\}$, except that the entries are quaternions and $A^*$ means **transpose and quaternionic conjugate**. Nothing in the definition is nonlinear: after replacing each quaternion by four real numbers, an element of $Sp(n)$ is an ordinary $4n\times4n$ real matrix satisfying ordinary polynomial equations.

The notation has one trap. This article is about the **compact** group $Sp(n)$, sometimes written $USp(2n)$. A different group, the real symplectic group $Sp(2n,\mathbb R)$, preserves a skew-symmetric bilinear form and is noncompact. The two are related, but they are not the same group.

\documentclass[border=6pt]{standalone}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\begin{document}
\begin{tikzpicture}[>=Latex,node distance=11mm and 17mm,
 box/.style={draw,rounded corners,align=center,inner sep=5pt},
 arr/.style={->,thick}]
 \node[box] (r) {$O(n)$\\real matrices\\$A^TA=I$};
 \node[box,right=of r] (c) {$U(n)$\\complex matrices\\$A^*A=I$};
 \node[box,right=of c] (h) {$Sp(n)$\\quaternionic matrices\\$A^*A=I$};
 \draw[arr] (r)--node[above,font=\scriptsize]{allow $i$}(c);
 \draw[arr] (c)--node[above,font=\scriptsize]{allow $j,k$}(h);
 \node[below=8mm of c,align=center,font=\small] {the same orthonormal-column idea over\\$\mathbb R$, $\mathbb C$, and $\mathbb H$};
\end{tikzpicture}
\end{document}

## First meet the quaternions

A quaternion is

$$q=a+bi+cj+dk,\qquad a,b,c,d\in\mathbb R,$$

with multiplication determined by

$$i^2=j^2=k^2=ijk=-1.$$

Thus $ij=k$ but $ji=-k$: multiplication is associative but not commutative. Conjugation and length are

$$\bar q=a-bi-cj-dk,\qquad |q|^2=q\bar q=a^2+b^2+c^2+d^2.$$

So $mathbb H$ is just $mathbb R^4$ as a real vector space, equipped with extra multiplication. The identity $|pq|=|p||q|$ is why quaternionic matrices can have a useful notion of orthogonality.

For a column $v=(q_1,\ldots,q_n)^T\in\mathbb H^n$, define

$$\langle v,w\rangle_{\mathbb H}=v^*w=\sum_{r=1}^n\bar q_r w_r,
\qquad \|v\|^2=\langle v,v\rangle_{\mathbb H}=\sum_r|q_r|^2.$$

The inner product is quaternion-valued. Its real part is the ordinary dot product on $mathbb R^{4n}$. Consequently $A^*A=I$ says exactly what it says for real orthogonal matrices: the columns of $A$ form an orthonormal basis, now over $mathbb H$.

## One quaternion as a real matrix

Left multiplication by $q=a+bi+cj+dk$ is a real-linear map $mathbb H\to\mathbb H$. In the basis $(1,i,j,k)$ its matrix is

$$L_q=
\begin{pmatrix}
a&-b&-c&-d\\
b&a&-d&c\\
c&d&a&-b\\
d&-c&b&a
\end{pmatrix}.$$

Direct multiplication gives $L_q^TL_q=|q|^2I_4$. Therefore every unit quaternion gives a $4\times4$ orthogonal matrix. This is already the whole group $Sp(1)$:

$$Sp(1)=\{q\in\mathbb H:|q|=1\}=S^3\cong SU(2).$$

There is a useful way to draw all four coordinates without pretending that $S^3$ fits in ordinary space. Write a unit quaternion as

$$q=\cos\theta+\mathbf u\sin\theta,$$

where $\mathbf u=u_1i+u_2j+u_3k$ is a unit imaginary quaternion. The sphere below contains the imaginary vector $(b,c,d)=\mathbf u\sin\theta$; the circular gauge on top carries the missing angle $\theta$ and hence the real coordinate $a=\cos\theta$. Together they specify the full quaternion. The faint back half of the sphere and the dashed part of the vector indicate depth. Drag horizontally to choose $\theta$ and vertically to turn the imaginary direction.

<p5sketch width="400" height="400">
    let quaternionGaugeAngle = 0.72;
    let quaternionAxisAzimuth = -0.65;
    let quaternionAxisElevation = 0.38;
    let quaternionGaugeDragging = false;

    function setup() {
      createCanvas(sketchWidth, sketchHeight);
      textFont('sans-serif');
    }

    function projectQuaternionPoint(x, y, z) {
      const yaw = -0.62, pitch = 0.34;
      const x1 = x*cos(yaw) - z*sin(yaw);
      const z1 = x*sin(yaw) + z*cos(yaw);
      return [x1, y*cos(pitch)-z1*sin(pitch), y*sin(pitch)+z1*cos(pitch)];
    }

    function drawQuaternionVectorArrow(x0, y0, x1, y1, colorValue) {
      stroke(colorValue); strokeWeight(3); line(x0,y0,x1,y1);
      const headingValue=atan2(y1-y0,x1-x0);
      noStroke(); fill(colorValue);
      push(); translate(x1,y1); rotate(headingValue);
      triangle(0,0,-10,-5,-10,5); pop();
    }

    function draw() {
      background(18,21,28);
      const cx=200, cy=244, radiusValue=112;

      // soft spherical body
      noStroke();
      for(let r=radiusValue;r>0;r-=3){
        const shadeValue=map(r,0,radiusValue,47,20);
        fill(shadeValue,shadeValue+5,shadeValue+13);
        circle(cx-0.12*(radiusValue-r),cy-0.12*(radiusValue-r),2*r);
      }
      noFill(); stroke(105,125,150); strokeWeight(1.3); circle(cx,cy,2*radiusValue);
      stroke(70,88,108); ellipse(cx,cy,2*radiusValue,55);
      drawingContext.setLineDash([5,5]);
      arc(cx,cy,2*radiusValue,55,PI,TWO_PI);
      drawingContext.setLineDash([]);

      // q = cos(theta) + u sin(theta), with u moved by the vertical drag
      const s=sin(quaternionGaugeAngle), a=cos(quaternionGaugeAngle);
      const ux=cos(quaternionAxisElevation)*cos(quaternionAxisAzimuth);
      const uy=sin(quaternionAxisElevation);
      const uz=cos(quaternionAxisElevation)*sin(quaternionAxisAzimuth);
      const projected=projectQuaternionPoint(ux*s,uy*s,uz*s);
      const vx=cx+radiusValue*projected[0];
      const vy=cy-radiusValue*projected[1];

      if(projected[2]<0){
        drawingContext.setLineDash([6,5]);
        drawQuaternionVectorArrow(cx,cy,vx,vy,'#f2bd57');
        drawingContext.setLineDash([]);
      } else {
        drawQuaternionVectorArrow(cx,cy,vx,vy,'#f2bd57');
      }
      noStroke(); fill(241,189,87); circle(vx,vy,7);

      // circular gauge for theta and the real coordinate a = cos(theta)
      const gx=200, gy=62, gr=38;
      noFill(); stroke(72,86,104); strokeWeight(8); circle(gx,gy,2*gr);
      stroke(91,203,222); strokeWeight(8); strokeCap(ROUND);
      arc(gx,gy,2*gr,2*gr,-HALF_PI,-HALF_PI+quaternionGaugeAngle);
      strokeCap(SQUARE);
      const needleAngle=-HALF_PI+quaternionGaugeAngle;
      stroke(230); strokeWeight(2); line(gx,gy,gx+27*cos(needleAngle),gy+27*sin(needleAngle));
      noStroke(); fill(235); circle(gx,gy,5);
      textAlign(CENTER); textSize(12); fill(210);
      text('θ',gx,gy+4); text('a = cos θ = '+a.toFixed(2),gx,116);

      textAlign(LEFT); textSize(12); fill(160);
      text('imaginary vector  (b, c, d)',16,382);
      fill(241,189,87); circle(177,378,6);
    }

    function mousePressed(){
      if(mouseX>=0&&mouseX<width&&mouseY>=0&&mouseY<height) quaternionGaugeDragging=true;
    }
    function mouseDragged(){
      if(!quaternionGaugeDragging) return;
      quaternionGaugeAngle=map(constrain(mouseX,0,width),0,width,0,TWO_PI);
      quaternionAxisElevation=map(constrain(mouseY,120,height),120,height,-HALF_PI,HALF_PI);
      quaternionAxisAzimuth+=0.012*(mouseX-pmouseX);
    }
    function mouseReleased(){ quaternionGaugeDragging=false; }
</p5sketch>

The sketch below shows the four real coordinate axes $1,i,j,k$ and their images under left multiplication by $q=\cos t+i\sin t$. What looks like one complex rotation is actually the same-angle rotation in two perpendicular planes: the $(1,i)$ plane and the $(j,k)$ plane. It runs automatically; drag horizontally to choose $t$ yourself.

<p5sketch width="620" height="330">
    let quaternionAngle = 0;
    let angleDragging = false;

    function setup() {
      createCanvas(sketchWidth, sketchHeight);
      textFont('sans-serif');
    }

    function drawQuaternionArrow(x0, y0, x1, y1, col, label) {
      stroke(col); strokeWeight(2.5); fill(col);
      line(x0, y0, x1, y1);
      const a = atan2(y1-y0, x1-x0);
      push(); translate(x1,y1); rotate(a);
      triangle(0,0,-9,-4,-9,4); pop();
      noStroke(); textSize(13); text(label,x1+6,y1-5);
    }

    function drawRotationPlane(cx, cy, title, names, phase) {
      stroke(75); strokeWeight(1); line(cx-105,cy,cx+105,cy); line(cx,cy-88,cx,cy+88);
      noStroke(); fill(205); textAlign(CENTER); textSize(14); text(title,cx,cy-105);
      const r=72, a=quaternionAngle+phase;
      drawQuaternionArrow(cx,cy,cx+r*cos(a),cy-r*sin(a),'#ffb347',names[0]);
      drawQuaternionArrow(cx,cy,cx-r*sin(a),cy-r*cos(a),'#62c7df',names[1]);
    }

    function draw() {
      background(18,21,28);
      if (!angleDragging) quaternionAngle += 0.012;
      fill(230); noStroke(); textAlign(LEFT); textSize(13);
      text('q = cos t + i sin t     t = '+(quaternionAngle%TWO_PI).toFixed(2),18,24);
      drawRotationPlane(160,182,'span{1, i}',['q·1','q·i'],0);
      drawRotationPlane(460,182,'span{j, k}',['q·j','q·k'],0);
      fill(155); textAlign(CENTER); text('orange and blue remain perpendicular and unit length',width/2,316);
    }

    function mousePressed(){ if(mouseX>=0&&mouseX<width&&mouseY>=0&&mouseY<height) angleDragging=true; }
    function mouseDragged(){ if(angleDragging) quaternionAngle=map(mouseX,0,width,0,TWO_PI); }
    function mouseReleased(){ angleDragging=false; }
</p5sketch>

## From quaternionic matrices to real matrices

Replace each entry $a_{rs}$ of a quaternionic matrix by its $4\times4$ block $L_{a_{rs}}$. This turns $A\in M_n(\mathbb H)$ into an ordinary real $4n\times4n$ matrix. Under this replacement,

$$A^*A=I \quad\Longrightarrow\quad A_{\mathbb R}^TA_{\mathbb R}=I_{4n}.$$

Hence $Sp(n)$ is a subgroup of $O(4n)$. In fact its determinant is always $+1$, so

$$Sp(n)\subset SO(4n).$$

But most rotations of $mathbb R^{4n}$ are not in $Sp(n)$. A quaternionic-linear map must also respect right multiplication by $i,j,k$. Regard these as three fixed real-linear maps $I,J,K$ on $mathbb R^{4n}$. They satisfy

$$I^2=J^2=K^2=-I_{4n},\qquad IJ=K=-JI.$$

Then there is an entirely real definition:

$$Sp(n)=\{A\in SO(4n):AI=IA,\ AJ=JA,\ AK=KA\}.$$

Because $K=IJ$, commuting with $I$ and $J$ already forces commuting with $K$. This characterization says: $Sp(n)$ consists of rotations that do not disturb the three linked quarter-turn structures carried by quaternionic space.

In the next sketch, a point represents a vector in one quaternionic coordinate. The colored arrows represent $v$, $Iv$, $Jv$, and $Kv$ after a two-dimensional projection. Move the pointer to choose $v$. The visible lengths can change under projection, but the algebraic rules tying the four vectors together do not.

<p5sketch width="620" height="350">
    function setup(){ createCanvas(sketchWidth,sketchHeight); textFont('sans-serif'); }
    function arr(x0,y0,x1,y1,c,s){
      stroke(c); strokeWeight(3); line(x0,y0,x1,y1);
      const a=atan2(y1-y0,x1-x0); fill(c); noStroke();
      push(); translate(x1,y1); rotate(a); triangle(0,0,-9,-4,-9,4); pop();
      textSize(14); text(s,x1+5,y1-5);
    }
    function draw(){
      background(19,22,29); const cx=width/2, cy=height/2+8;
      stroke(65); strokeWeight(1); line(25,cy,width-25,cy); line(cx,40,cx,height-25);
      const x=constrain(mouseX-cx,-115,115), y=constrain(mouseY-cy,-115,115);
      // A readable 2D shadow of four 4D vectors. Two independent planes are offset.
      arr(cx,cy,cx+x,cy+y,'#f1c75b','v');
      arr(cx,cy,cx-y,cy+x,'#59c7df','Iv');
      arr(cx,cy,cx+0.72*x-0.45*y,cy-0.45*x-0.72*y,'#ec7892','Jv');
      arr(cx,cy,cx+0.45*x+0.72*y,cy+0.72*x-0.45*y,'#91d27b','Kv');
      noStroke(); fill(225); textAlign(LEFT); textSize(14);
      text('move the pointer: one vector generates a quaternionic frame',18,24);
      fill(155); textSize(12); text('a 2D shadow of a 4D configuration',18,height-14);
    }
</p5sketch>

The sketch is a projection, not a proof. The proof is matrix multiplication: $I,J,K$ are orthogonal, square to $-I$, anticommute, and send every nonzero $v$ to four mutually orthogonal real vectors $v,Iv,Jv,Kv$. Repeating this in $n$ quaternionic coordinate directions organizes $mathbb R^{4n}$ into $n$ such frames.

## A second definition inside complex matrices

Every quaternion can be written uniquely as $q=z+w j$ with $z,w\in\mathbb C$. The rule

$$z+w j\longmapsto
\begin{pmatrix}z&w\\-\bar w&\bar z\end{pmatrix}$$

turns quaternion multiplication into ordinary complex matrix multiplication. Applying it entry by entry embeds an $n\times n$ quaternionic matrix into a $2n\times2n$ complex matrix.

Let

$$\Omega=\begin{pmatrix}0&I_n\\-I_n&0\end{pmatrix}.$$

Then the compact symplectic group can also be written

$$Sp(n)=\{A\in U(2n):A^T\Omega A=\Omega\}=U(2n)\cap Sp(2n,\mathbb C).$$

This explains both words in “compact symplectic.” The equation $A^*A=I$ is unitary and makes the group compact; the equation $A^T\Omega A=\Omega$ preserves a complex skew form and makes it symplectic. It also explains the alternative notation $USp(2n)$.

Be careful about the two transposes: $A^*$ includes complex conjugation and controls length, while $A^T$ does not and controls the skew form.

\documentclass[border=6pt]{standalone}
\usepackage{amssymb}
\usepackage{tikz}
\usetikzlibrary{arrows.meta,positioning}
\begin{document}
\begin{tikzpicture}[>=Latex,node distance=12mm,
 n/.style={draw,rounded corners,align=center,minimum width=34mm,minimum height=10mm}]
 \node[n] (h) {$A\in M_n(\mathbb H)$\\$A^*A=I$};
 \node[n,right=20mm of h] (c) {$A\in M_{2n}(\mathbb C)$\\$A^*A=I$, $A^T\Omega A=\Omega$};
 \node[n,below=of c] (r) {$A\in M_{4n}(\mathbb R)$\\$A^TA=I$, $[A,I]=[A,J]=0$};
 \draw[->,thick] (h)--node[above,font=\scriptsize]{$q=z+wj$}(c);
 \draw[->,thick] (c)--node[right,font=\scriptsize]{split real/imaginary parts}(r);
 \draw[->,thick,bend right=25] (h) to node[left,font=\scriptsize]{four real coordinates}(r);
\end{tikzpicture}
\end{document}

## Why it is a Lie group

A **Lie group** is simultaneously a group and a smooth space, with smooth multiplication and inversion. Here the entries of $A$ provide coordinates in a real Euclidean space, and $A^*A=I$ is a finite list of polynomial equations. The solution set is closed and bounded, hence compact. It is also a smooth manifold; its tangent space reveals the local shape.

Take a smooth path $A(t)$ in $Sp(n)$ with $A(0)=I$ and velocity $X=A'(0)$. Differentiate $A(t)^*A(t)=I$ at $t=0$:

$$X^*+X=0.$$

Thus the Lie algebra is

$$\mathfrak{sp}(n)=\{X\in M_n(\mathbb H):X^*=-X\}.$$

These are the quaternionic skew-Hermitian matrices. Count real parameters:

  * each diagonal entry must be purely imaginary, giving $3n$ parameters;
  * each pair above/below the diagonal is determined by one arbitrary quaternion, giving $4\binom n2$ parameters.

Therefore

$$\dim Sp(n)=3n+4\binom n2=n(2n+1).$$

For comparison, $dim O(n)=n(n-1)/2$ and $dim U(n)=n^2$. The dimension is the number of independent infinitesimal motions near the identity—not the size of the matrices representing them.

Exponentiating a tangent matrix produces a group element:

$$X\in\mathfrak{sp}(n)\quad\Longrightarrow\quad e^X\in Sp(n).$$

Indeed $(e^X)^*=e^{X^*}=e^{-X}=(e^X)^{-1}$. This is the same matrix exponential familiar from linear differential equations.

## Building an orthonormal quaternionic basis

$Sp(n)$ acts on the unit sphere $S^{4n-1}\subset\mathbb H^n$. It acts **transitively**: given unit vectors $v$ and $w$, quaternionic Gram–Schmidt extends each to an orthonormal basis, and the matrix sending one basis to the other lies in $Sp(n)$.

Once the first column is chosen, the remaining columns form an element of $Sp(n-1)$ on its orthogonal complement. This gives a bundle

$$Sp(n-1)\longrightarrow Sp(n)\longrightarrow S^{4n-1}$$

and the homogeneous-space identity

$$S^{4n-1}\cong Sp(n)/Sp(n-1).$$

The dimension check is reassuring:

$$\dim Sp(n)-\dim Sp(n-1)=4n-1=\dim S^{4n-1}.$$

The sketch visualizes the real analogue of this column-by-column construction. Drag the yellow first column; the blue column is forced to stay perpendicular. Quaternionic Gram–Schmidt uses the same projection-and-subtraction recipe, but one chosen quaternionic column accounts for four linked real directions.

<p5sketch width="620" height="300">
    function setup(){ createCanvas(sketchWidth,sketchHeight); textFont('sans-serif'); }
    function arrow(cx,cy,x,y,c,label){
      stroke(c); strokeWeight(4); line(cx,cy,x,y); const a=atan2(y-cy,x-cx);
      noStroke(); fill(c); push(); translate(x,y); rotate(a); triangle(0,0,-10,-5,-10,5); pop();
      textSize(14); text(label,x+7,y-4);
    }
    function draw(){
      background(19,22,29); const cx=205,cy=157;
      let x=constrain(mouseX-cx,-105,105), y=constrain(mouseY-cy,-105,105);
      let d=sqrt(x*x+y*y); if(d<12){x=82;y=-45;d=sqrt(x*x+y*y);}
      x=112*x/d; y=112*y/d;
      stroke(65); strokeWeight(1); noFill(); circle(cx,cy,224);
      arrow(cx,cy,cx+x,cy+y,'#f1c75b','first column v');
      arrow(cx,cy,cx-y,cy+x,'#59c7df','forced perpendicular');
      noStroke(); fill(230); textAlign(LEFT); textSize(15);
      text('orthonormal columns move together',18,25);
      fill(190); textSize(13);
      text('choose v on the unit circle',385,105);
      text('the remaining choice lives',385,137);
      text('in its orthogonal complement',385,158);
      fill('#f1c75b'); text('drag inside the sketch',385,205);
    }
</p5sketch>

Iterating the bundle suggests—but does not by itself prove—that $Sp(n)$ is connected and simply connected. Both statements are true. Its Lie algebra belongs to the $C_n$ family, one of the four infinite classical families $A_n,B_n,C_n,D_n$.

## Small cases

### $Sp(1)$

As already seen,

$$Sp(1)=S^3=\{a+bi+cj+dk:a^2+b^2+c^2+d^2=1\}\cong SU(2).$$

It double-covers $SO(3)$. A unit quaternion $q$ rotates an imaginary quaternion $x=xi+yj+zk$ by

$$x\longmapsto qxq^{-1}.$$

The quaternions $q$ and $-q$ give the same rotation, which accounts for the two-to-one map.

### $Sp(2)$

$Sp(2)$ has dimension $2(5)=10$ and acts on $mathbb H^2\cong\mathbb R^8$. There is an exceptional isomorphism

$$Sp(2)/\{\pm I\}\cong SO(5),$$

or, at the simply connected level, $Sp(2)\cong Spin(5)$. This is a low-dimensional coincidence, not a pattern continuing for all $n$.

## Quaternionic projective space

Real projective space identifies nonzero real vectors that differ by a real scale. Complex projective space does the same with complex scales. Quaternionic projective space is

$$\mathbb HP^{n-1}=(\mathbb H^n\setminus\{0\})/\mathbb H^\times.$$

After normalizing length, only unit-quaternion scalars remain, so there is a quaternionic Hopf fibration

$$Sp(1)=S^3\longrightarrow S^{4n-1}\longrightarrow\mathbb HP^{n-1}.$$

The group $Sp(n)$ moves quaternionic lines transitively. The line through the first basis vector is fixed by $Sp(1)\times Sp(n-1)$, giving

$$\mathbb HP^{n-1}\cong \frac{Sp(n)}{Sp(1)\times Sp(n-1)}.$$

For $n=2$, this says $\mathbb HP^1\cong S^4$. It is the quaternionic sibling of $\mathbb CP^1\cong S^2$ and connects $Sp(n)$ directly to the [[wiki:hopf-fibration|Hopf fibration]].

## Rank, maximal torus, and roots

Commutativity returns inside a useful subgroup. Choose the same complex plane $\{a+bi\}\subset\mathbb H$ in every diagonal entry. Then

$$T=\left\{\operatorname{diag}(e^{i\theta_1},\ldots,e^{i\theta_n})\right\}\cong(S^1)^n$$

is a **maximal torus**, so $Sp(n)$ has rank $n$. Every element of $Sp(n)$ is conjugate to an element of this torus: after a suitable quaternionic change of orthonormal basis, its essential motion is described by $n$ angles.

Relative to angle coordinates $e_1,\ldots,e_n$, the roots are

$$\pm e_r\pm e_s\quad(r\ne s),\qquad \pm2e_r.$$

This is the root system $C_n$, which is why the Lie algebra is written $\mathfrak{sp}(n)$ and classified as type $C_n$. The short roots mix two quaternionic coordinate directions; the long roots act within one. Counting roots gives $2n^2$, and adding the $n$-dimensional torus recovers

$$2n^2+n=n(2n+1)=\dim Sp(n).$$

## What “preserved” means

The definitions can now be read as different views of one object:

^ View of the vector space ^ Matrices ^ What is preserved ^
| $\mathbb H^n$ | $n\times n$ quaternionic | quaternionic inner product $v^*w$ |
| $\mathbb C^{2n}$ | $2n\times2n$ complex | Hermitian length and $v^T\Omega w$ |
| $\mathbb R^{4n}$ | $4n\times4n$ real | Euclidean length and the structures $I,J,K$ |

The quaternionic row is the shortest definition. The complex row explains the word *symplectic*. The real row makes clear that the group is accessible using only real vectors and matrices: it is a specially constrained family of rotations in $4n$ dimensions.

## A practical recognition checklist

Given a candidate transformation, any one of these routes is enough:

  - **Quaternionic form:** verify $A^*A=I$.
  - **Complex form:** verify both $A^*A=I$ and $A^T\Omega A=\Omega$.
  - **Real form:** verify $A^TA=I$ and that $A$ commutes with the fixed matrices $I$ and $J$.
  - **Infinitesimal form:** for a generator $X$, verify $X^*=-X$; then $e^{tX}\in Sp(n)$ for every real $t$.

The mental model to keep is simple: $O(n)$ moves real orthonormal frames, $U(n)$ moves complex orthonormal frames, and $Sp(n)$ moves quaternionic orthonormal frames. Each extra number system adds structure that a transformation must respect—and creates a richer group of coordinated rotations when viewed over the real numbers.