Within the context of the Wedderburn-Malcev theorem a radical complement exists and all
complements are conjugated. The main topics of this work are to analyze the Determination of a
(all) radical complements the representation of an element as the sum of a nilpotent and fully
separable element and the compatibility of the Wedderburn-Malcev theorem with derived
structures. Answers are presented in details for commutative and solvable associative algebras.
Within the analysis the set of fully-separable elements and the generalized Jordan
decomposition are of special interest. We provide examples based on generalized quaternion
algebras group algebras and algebras of traingular matrices over a field. The results (and
also the theorem of Wedderburn-Malcev and Taft) are transferred to non-unitary algebras by
using the star-composition and the adjunction of an unit. Within the App endix we present
proofs for the Wedderburn-Malcev theorem for unitary algebras for Taft's theorem on
G-invariant radical complements for unitary algebras and for a theorem of Bauer concerning
solvable unit groups of associative algebras.
