Let $\mathcal{G} = \mathbb{M}_n(\mathbb{C})$ be an $n$-by-$n$ matrix algebra over complex numbers. Next let $\mathcal{A} \cong \mathbb{M}_d(\mathbb{C})$ be a subalgebra of $\mathcal{G}$ and assume $d$ divides $n$. Then is it true that there exists another subalgebra of $\mathcal{G}$ (let's call it $\mathcal{B}$) such that

1) $\mathcal{G} \cong \mathcal{A} \otimes \mathcal{B}$.

2) Every element in $\mathcal{A}$ commutes with every element in $\mathcal{B}$.

3) The intersection of $\mathcal{A}$ and $\mathcal{B}$ are multiples of the identity.

If it is not true, what are the necessary conditions?