# Flat Lie algebras

I would like to review a classical result by Milnor, Curvatures of left invariant metrics on Lie groups. J., Adv. Math. 21 (1976), no. 3, 293-329.

Theorem 1.5 (page 298). A Lie group with left invariant metric is flat if and only if the associated Lie algebra splits as an orthogonal direct sum $\mathfrak{b}\oplus\mathfrak{u}$ where $\mathfrak{b}$ is a commutative sub algebra, $\mathfrak{u}$ is a commutative ideal, and where the linear transformation $\mathrm{ad}(b)$ is skew-adjoint for every $b\in\mathfrak{b}$.

In the proof (page 326), Milnor uses two geometrical results: Cauchy-Hadamard theorem and Myers theorem.

My question is as follows:

Is there any algebraic proof of Milnor's theorem? May one uses Chevalley-Eilenberg cohomology or such algebraic tools to give a 'new' proof to the result aboe?

Although, I proved (with Pr. Boucetta) an improved version of Milnor's theorem. See http://arxiv.org/abs/0906.2887 proposition 2.1 and 2.2, page 5.

• Hello, sir. Sorry for comment such an old question. I've read a little the paper you mentioned. If I understand correctly, you've proven an improved version of Milnor's theorem without using Milnor's theorem? – Ale Tolcachier Jan 8 at 22:37
• It seems that now there is an '' algebraic proof'' mathem.pub.ro/dgds/v16/D16-ch-733.pdf (I googled "Algebraic proof of Milnor flat lie algebras" or something like that, I hope the link won't broke) – Ale Tolcachier Jan 15 at 13:47