Remark 1

Tag 01BB, permanent. Cite it as [Bourbaki, Tag 01BB].

Lie Groups and Lie Algebras, chapter IX, § 3. Compact forms of complex semi-simple Lie algebras

We have $\mathfrak{g}\mathfrak{l}(V) =\mathfrak{s}\mathfrak{l}(V)\times \mathbf{C}.1_V,\mathfrak{u}(\Phi ) =\mathfrak{s}\mathfrak{u}(\Phi )\times \mathbf{R}.i1_V$ ; the compact real forms of $\mathfrak{g}\mathfrak{l}(V)$ are the $\mathfrak{s}\mathfrak{u}(\Phi )\times \mathbf{R}.\alpha 1_V,\alpha \in \mathbf{C}^*$ .

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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