Definition 1

Tag 0A67, permanent. Cite it as [Bourbaki, Tag 0A67].

Lie Groups and Lie Algebras, chapter II, § 6. The Hausdorff series

The element $H = U \circledast V = \log(\exp U \cdot \exp V)$ (no. 2) of the Lie algebra $\hat{L}_\mathbf{Q}(\{U, V\})$ is called the Hausdorff series in the indeterminates U and V.

We denote by $H_n$ (resp. $H_{r,s}$ ) the homogeneous component of H of total degree n (resp. multidegree $(r, s)$ ). Then

H = \sum_{n \geq 0} H_n = \sum_{r, s \geq 0} H_{r,s}, \quad H_n = \sum_{r+s=n \atop r, s \geq 0} H_{r,s}.

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

Cited by

§ 7. Convergence of the Hausdorff series (real or complex case)
§ 8. Convergence of the Hausdorff series (ultrametric case)
Lemma 2, § 8
§ 4. Passage from Lie algebras to Lie groups

Cites

Nothing.