Relate Direct Sum and Kronecker Product of Matrices

Is there a general relation between the direct sum of matrices $A \oplus B = \mathrm{diag}(A,B)$, yielding a block diagonal matrix, and the Kronecker product $(A \otimes B)_{ij} = a_{ij} B$?

For instance i know that $I_{2 \times 2} \otimes B = B \oplus B$. I was wondering if there is a more general way to express one using the other. Especially I would be interested in an expression for $B \otimes I_{2 \times 2}$ involving a direct sum.

Thanks!

• I am almost sure there is no more general relationship. – Jean Marie Dec 20 '16 at 23:39