Corollary

Tag 0A93, permanent. Cite it as [Bourbaki, Tag 0A93].

Lie Groups and Lie Algebras, chapter III, § 2. Group of tangent vectors to a Lie group

Let $G_1, \ldots, G_n$ be Lie groups. The canonical isomorphism of the manifold $T(G_1 \times \cdots \times G_n)$ onto the manifold $T(G_1) \times \cdots \times T(G_n)$ is a Lie group isomorphism.

$\mathrm{pr}_i$ is a morphism of $G_1 \times \cdots \times G_n$ into $G_i$ and hence $T(\mathrm{pr}_i)$ is a morphism of $T(G_1 \times \cdots \times G_n)$ into $T(G_i)$ .

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

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