Corollary 2

Tag 0AAX, permanent. Cite it as [Bourbaki, Tag 0AAX].

Lie Groups and Lie Algebras, chapter III, § 3. Passage from a Lie group to its Lie algebra

*Let E be a complex (resp. real) Hilbert space and U the unitary group of E. Then U is a real subgroup of $\mathbf{GL}(E)$ and $L(U)$ is the set of $x \in \mathscr{L}(E)$ such that $x^* = -x$ .

We apply Proposition 37 with $A = \mathscr{L}(E)$ considered as an algebra over $\mathbf{R}$ and $x^t = x^*$ .*

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

Cited by

§ 3. Compact forms of complex semi-simple Lie algebras

Cites

Proposition 37