During the author's doctorate time at the Christian-Albrechts-Universitat to Kiel Salvatore
Siciliano gave a stimulating talk in the upper seminar algebra theory about Cartan subalgebras
in Lie algebra associates to associative algebra. This talk was the incentive for the author to
analyze maximal nilpotent substructures of the Lie algebra associated to associative algebras.
In the present work Siciliano's theory about Cartan subalgebras is worked off and expanded to
different special associative algebra classes. In addition a second maximal nilpotent
substructure is analyzed: the nilradical. Within this analysis the main focus is to describe
these substructure with the associative structure of the underlying algebra. This is
successfully realized in this work. Numerous examples (like group algebras and Solomon (Tits-)
algebras) illustrate the results to the reader. Within the numerous exercises these results can
be applied by the reader to get a deeper insight in this theory.
