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?