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 $<v>$ is a subspace which is invariant under $L$. Decompose $V$ as $V=<v> oplus V’$. 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.