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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.

Extension of the Product of a Post-Lie Algebra and Application to the SISO Feedback Transformation Group