Zero action on a Lie algebra quotient submodule

Question

Let $L $ be a Lie algebra and $I$ be an ideal in $L$. If $M,N$ are $L$-module with $N\subset M$.

If $M/N$ is an $L/I$-module and $K/N$ is am $L/I$-submodule of $M/N$ with $(L/I)\cdot (K/N)=0$. Does this imply that $L.K\subset N$ regardless how $I$ acts on $K$?