Is there an ideal $J$ with $Ann(J)=0$ such that $J\subseteq I_1\cap\cdots\cap I_n$?

Question

Let $L$ be a Lie algebra and $I_1,\dots,I_n$ ideals in $L$ with $Ann(I_k)=0$ for all $k=1,\dots,n$. Of course, $I_1\cap\cdots\cap I_n\neq0$. I would like to know if there exists an ideal $J\subseteq I_1\cap\cdots\cap I_n$ such that $Ann(J)=0$.

I'll appreciate any suggestion.