Proposition 6

Let

be an extension of $b$ by $a$ and $n$ its kernel.

(a) If there exists a subalgebra m of g supplementary to n in g, the restriction of μ to m is an isomorphism of m onto b. If ν denotes the inverse isomorphism of this restriction, ν is a homomorphism of b into g and μ ∘ ν is the identity automorphism of b.

(b) Conversely, if there exists a homomorphism ν of b into g such that μ ∘ ν is the identity automorphism of b, then ν(b) is a supplementary subalgebra of n in g.

The assertions of (a) are immediate. On the other hand, let ν be a homomorphism of b into g such that μ ∘ ν is the identity automorphism of b. Then ν(b) is a subalgebra of g and g is the direct sum of ν(b) and $\mu^{-1}(0) = n$ (Algebra, Chapter VIII, § 1, no. 1).