# lie algebras – Proof of Lie’s theorem using theorem 4.1 in Humphreys

I’m studying Humphreys’ book ‘Introduction to Lie Algebras and Representation Theory’
First here is theorem 4.1.

Theorem. Let $$L$$ be a solvable subalgebra of $$mathfrak gl(V)$$, $$V$$ finite dimensional. if $$V neq 0$$, then $$V$$ contains a common eigenvector for all the endomorphisms in $$L$$.

And here is Corollary A.

Coroallary A. (Lie’s Theorem). Let $$L$$ be a solvable subalgebra of $$mathfrak gl(V)$$, dim$$V=n lt infty$$. Then $$L$$ stabilizes some flag in $$V$$.

And the book states that we can prove Cor.A. by using the above theorem along with induction on dim$$V$$. Here is my attempt.

If dim$$V$$=1, then the only flag is $$0 subset V$$ and it is stabilized clearly by $$L$$. Now assume dim$$V=n$$. By the main theorem, there exists $$v in V$$ such that $$x.v=lambda(x)v$$ for all $$xin L$$. Then $$$$ is a subspace which is invariant under $$L$$. Decompose $$V$$ as $$V= oplus V’$$. Let $$L’=Lcap mathfrak gl(V’)$$. Clearly, $$L’$$ is solvable in $$mathfrak gl(V’)$$. Since dim$$V’lt$$ dim$$V$$, by induction hypothesis, there is a flag of $$V’$$ stabilized by $$L’$$.

Here is where I stucked. How can I extend that flag of $$V’$$ stabilized by $$L’$$ to a flag of $$V$$ stabilized by L? I found other questions about this problem, but I still don’t get it.

Thanks for any help in advance.