I have read John Baez on symplectic but not understood the relation between the real symplectic group $Sp(2n,\mathbb R)$ and the quaternion group $Sp(n)$. They seem to both have real dimension $n(2n+1)$ which I understand in the real case but not in the quaternion case. I know that there is a matrix representation of a quaternion as a $4 \times 4$ real orthogonal matrix or a $2 \times 2$ complex matrix so that the quaternion case would be isomorph to $Sp(4n,\mathbb R)$ and $Sp(2n,\mathbb C)$ respectively.
Another confusion is how a quaternion inner product decomposes into an orthogonal and symplectic case. This is clear to me in the complex case where the orthogonal structure is the real part and the imaginary part is the symplectic one. The unitary group keeps these two parts invariant seperatly. But in the quaternion case there are three symplectic forms for each of $i$, $j$ and $k$ imaginary parts. What is the equivalent to the unitary group?