Proposition 4

a) The compact real forms of the complex Lie algebra $\mathfrak{s}\mathfrak{l}(V)$ are the algebras $\mathfrak{s}\mathfrak{u}(\Phi )$, where $\Phi$ belongs to the set of separating positive hermitian forms on the complex vector space V.

b) The algebras $\mathfrak{u}(\Phi )$ are the compact real forms of $\mathfrak{g}\mathfrak{l}(V)$.

Let $\Phi$ be a separating positive hermitian form on V. For all $x\in \mathfrak{g}\mathfrak{l}$(V), put $\sigma (x) =-x^*$ (where $x^*$ is the adjoint of $x$ relative to $\Phi$ ). Then $\sigma$ satisfies conditions (2) of Prop. 1 of no. 1, so the set $\mathfrak{u}(\Phi )$ (resp. $\mathfrak{s}\mathfrak{u}(\Phi )$) of fixed points of $\sigma$ on $\mathfrak{g}\mathfrak{l}(V)$ (resp. $\mathfrak{s}\mathfrak{l}(V)$) is a compact real form of $\mathfrak{g}\mathfrak{l}(V)$ (resp. $\mathfrak{s}\mathfrak{l}(V)$). Since $\mathbf{G}\mathbf{L}(V)$ operates transitively on the set of separating positive hermitian forms on V (Algebra, Chap. IX) and on the set of compact real forms of $\mathfrak{s}\mathfrak{l}(V)$ (no. 3, Th. 1 and Chap. VIII, §13, no. 1 (VII)), Prop. 4 is proved.