Exercise 6, § 3

a) Show that the compact Lie group $\mathbf{U}(n,\mathbf{H})$ is connected and simply-connected (use Exerc. 2), and that its centre is $\{\pm I_n\}$.

b) Every almost simple connected compact Lie group of type $C_n(n\geq 3)$ is isomorphic to $\mathbf{U}(n,\mathbf{H})$ or $\mathbf{U}(n,\mathbf{H})/\{\pm I_n\}$.