A quotient module of a Lie algebra

Question

Let $L$ be a Lie algebra. If $A$ and $B$ are $L$-submodules of an $L$-module $V$, such that $A\subset B$ and $I\cdot B\subset A$ for some ideal $A$ in $L$.

I want to understand why this implies that $B/A$ is an $L/I$-module?

Answer 1

There are two things going on here:

• If $A$ and $B$ are $L$-modules then so is $B/A$. Do you know how $L$ acts on $B/A$?
• If $Q$ is an $L$-module, $I\trianglelefteq L$ an ideal and every element of $I$ acts as the zero map on $Q$, then $Q$ naturally becomes an $L/I$-module. Do you know how $L/I$ acts on $Q$?

After understanding both of these things, apply with $Q=B/A$.