  We describe the both post-and pre-Lie algebra g SISO associated to the affine
SISO feedback transformation group. We show that it is a member of a family of
post-Lie algebras associated to representations of a particular solvable Lie
algebra. We first construct the extension of the magmatic product of a post-Lie
algebra to its enveloping algebra, which allows to describe free post-Lie
algebras and is widely used to obtain the enveloping of g SISO and its dual.
