Let $K$ be a number field, let $L⊆K$ be CM field. $A/K$ be an Abelian variety over $K$.
$A/K$ is said to have CM by $L$ if there is embedding $L⊆End_K(A) \otimes \Bbb{Q}$.
But if we adapt this definition, $L$ need not to be unique, I think. But I think it's weird to say if both $L$ and $L⊆L'$ satisfies the condition above, $A/K$ has CM by $L$. How can I eliminate this weirdness and get uniqueness of $L$ ?