Corollary

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

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

Let G be a Lie group. Let $T_e(G)$ be given its unique commutative Lie algebra structure. Let $\tau$ be an adjoint representation of $L(G)$ . Then the Lie algebra of $T(G)$ is the semi-direct product of $L(G)$ by $T_e(G)$ defined by $\tau$ . In other words, for $x, x'$ in $L(G)$ and $y, y'$ in $T_e(G)$ ,

[(x, y), (x', y')] = ([x, x'], [x, y'] + [y, x'])

(where the bracket on the left is evaluated in $L(T(G))$ and the brackets on the right in $L(G)$ ).

This follows from Proposition 48 and Proposition 6 of § 2, no. 2.

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

Cited by

Nothing.

Cites

Proposition 6 of § 2, no. 2
Proposition 48