Let $A$ be $C^{\ast}-$algebra and $I$ be closed subspace of $A$.
Suppose span$\{ab^*c-cb^*a: a,c \in A, b\in I \} \subset I$, can we reduce that $I$ is Lie ideal i.e. span$\{ab-ba: a\in I, b\in A\} \subset I$ or these ideals are called something else?
I cannot see any counterexample. Approximate identities also might be useful here. Any ideas?