Corollary 3

Let $\mathfrak{a}_1$ and $\mathfrak{a}_2$ be two complex semi-simple Lie algebras. The compact real forms of $\mathfrak{a}_1\times \mathfrak{a}_2$ are the products $\mathfrak{g}_1\times \mathfrak{g}_2$, where, for $i= 1,2,\mathfrak{g}_i$ is a compact real form of $\mathfrak{a}_i$.

Indeed, there exists a compact real form $\mathfrak{g}_1$ (resp. $\mathfrak{g}_2$) of $\mathfrak{a}_1$ (resp. $\mathfrak{a}_2$); then $\mathfrak{g}_1\times \mathfrak{g}_2$ is a compact real form of $\mathfrak{a}_1\times \mathfrak{a}_2$. The corollary now follows from Th. $1b)$, applied to $\mathfrak{a}_1,\mathfrak{a}_2$ and $\mathfrak{a}_1\times \mathfrak{a}_2$.

Note that it follows from Cor. 3 above that a compact real Lie algebra $\mathfrak{g}$ is simple if and only if the complex Lie algebra $\mathfrak{g}_{(\mathbf{C})}$ is simple. We say that $\mathfrak{g}$ is of type $A_n$, or $B_n,. .$., if $\mathfrak{g}_{(\mathbf{C})}$ is of type $A_n$, or $B_n,. .$. (Chap. VIII, §2, no. 2). By Cor. 1 above, two compact simple real Lie algebras are isomorphic if and only if they are of the same type.

Let G be an almost simple connected compact Lie group (Chap. III, §9, no. 8, Def. 3). We say that G is of type $A_n$, or $B_n,. .$., if its Lie algebra is of type $A_n$, or $B_n,. .$.. Two simply-connected almost simple compact Lie groups are isomorphic if and only if they are of the same type.