This paper considers finite-dimensional, linear time-varying dynamical systems (LDS) of the form $\dot {\bf x} = {\bf A}(t){\bf x}$, ${\bf x}(t_0 ) = {\bf x}_0 $. Such systems are said to be semiproper (self-commuting) if for all t, $\tau $, ${\bf A}(t){\bf A}(\tau ) = {\bf A}(\tau ){\bf A}(t)$. Using some recent results for obtaining explicit solutions for semiproper systems (see [26]–[29], [34], [35]), the family of Lyapunov reducible systems is expanded to include those systems that can be reduced to semiproper ones via what will be called D-similarity transformations. Within this new framework is defined the notion of primary D-similarity transformations, and every LDS that is “well defined” in a certain sense is proved reducible by a finite sequence of primaryD-similarity transformations. The paper also presents an explicit technique for constructing such transformations for LDS with virtually triangular ${\bf A}(t)$ (i.e., ${\bf A}(t) = {\bf L}{\bf T}(t){\bf L}^{ - 1} $ for some nonsingular constant...