About extreme case on complex interpolation

Question

I'm trying to prove some equality of spaces via complex interpolation with the usual Calderon functor $[,]_\theta$. If $(E_0,E_1)$ is a compatible couple, it is known that $$[E_0,E_1]_j, j=0,1,$$ is a closed subspace of $E_j$ with coincidence of norms in $[E_0,E_1]_j$. What is known about the equality $[E_0,E_1]_j=E_j$? In my particular case we have that $E_1$ is continuosly embedded in $E_0$, i suspect that in this case the equality $[E_0,E_1]_1=E_1$ could be achieved. I think that's true since $[E_0,E_1]_j$ is an intermediate space then $$E_1=E_0\cap E_1\hookrightarrow[E_0,E_1]_1 \hookrightarrow E_0+E_1=E_0,$$ but also $[E_0,E_1]_1\hookrightarrow E_1$, and hence $[E_0,E_1]=E_1$ with equivalence of the norms. Thanks in advance