On the divisors of infinitely divisible distributions admitting a Cartesian product representation

Let $n$-dimensional ($n\ge 2$) infinitely divisible distribution $P$ admits a representation in the form of Cartesian product of one-dimensional distributions. Let $P$ be also a convolution of two $n$-dimensional distributions $Q$ and $S$. We study the conditions under which the distributions $Q$ and $S$ must be the Cartesian products too.