  We prove that the convolution of a selfdecomposable distribution with its
background driving law is again selfdecomposable if and only if the background
driving law is s-selfdecomposable. We will refer to this as the
\textit{factorization property} of a selfdecomposable distribution; let $L^f$
denote the set of all these distributions. The algebraic structure and various
characterizations of $L^f$ are studied. Some examples are discussed, the most
interesting one being given by the L\'evy stochastic area integral. A nested
family of subclasses $L^{f}_n, n\ge 0,$ (or a filtration) of the class $L^f$ is
given.
