Exercise 3, § 3

a) Show that the Lie algebra of the real Lie group $\mathbf{U}(n,\mathbf{H})$ is the set of matrices $x\in M_n(\mathbf{H})$ such that $^t\overline{x}=-x$, with the bracket $[x, y] =xy-yx$. Denote it by $\mathfrak{u}(n,\mathbf{H})$.

b) Identify $\mathbf{C}$ with the subfield $\mathbf{R}(i)$ of $\mathbf{H}$, and $\mathbf{C}^{2n}$ with $\mathbf{H}^n$ by the isomorphism $(z_1, . . . , z_{2n}) \rightarrow (z_1+jz_{n+1}, . . . , z_n+jz_{2n})$. Prove the equality $\mathbf{U}(n,\mathbf{H}) =\mathbf{U}(2n,\mathbf{C})\cap \mathbf{S}\mathbf{p}(2n,\mathbf{C})$.

c) Deduce from b) that every compact simple (real) Lie algebra of type $C_n$ is isomorphic to $\mathfrak{u}(n,\mathbf{H})$.