# functional analysis – Trying to understand definition of Lie ideal for C*-algebras

Let $$A$$ be a $$C^*$$-algebra. A sub space $$I$$ of $$A$$ is called Lie ideal of A if $$[I,A]= IA-AI subset I$$

Since I contains $$0$$, isn’t it this definition equivalent to definition of two sided ideal of $$C^*$$-algerba?

Most probably I’m missing something in the definition of Lie ideal. Any ideas?