Let $H$ and $K$ be Hilbert spaces and $V\subset B(H,K)$ be a ternary ring of operators i.e. $xy^*z \in V$ for all $x,y,z \in V$. Let $I$ be a closed subspace of $V$. $I$ is called a ternary Lie ideal of $V$ provided $\operatorname{span}\{ab^*c-cb^*a: a,c \in V, b\in I \} \subset I$. One can check that every self-adjoint closed ternary Lie ideal of a $C^{\ast}$-algebra is actually a Lie ideal.
Does there exist an example of a ternary Lie ideal of a $C^{\ast}$-algebra which is not a Lie ideal?
I cannot see any example. Any ideas?