  We introduce a method for obtaining new classes of free divisors from
representations $V$ of connected linear algebraic groups $G$ where
$\dim(G)=\dim(V)$, with $V$ having an open orbit. We give sufficient conditions
that the complement of this open orbit, the "exceptional orbit variety", is a
free divisor (or a slightly weaker free* divisor) for "block representations"
of both solvable groups and extensions of reductive groups by them. These are
representations for which the matrix defined from a basis of associated
"representation vector fields" on $V$ has block triangular form, with blocks
satisfying certain nonsingularity conditions.
  For towers of Lie groups and representations this yields a tower of free
divisors, successively obtained by adjoining varieties of singular matrices.
This applies to solvable groups which give classical Cholesky-type
factorization, and a modified form of it, on spaces of $m \times m$ symmetric,
skew-symmetric or general matrices. For skew-symmetric matrices, it further
extends to representations of nonlinear infinite dimensional solvable Lie
algebras.
