The present work concerns certain aspects of Lie algebras, Lie triple systems andJordan triple systems. In particular, the following aspects are studied in detail: Liealgebras, Lie triple systems, Jordan algebras, Jordan triple systems and a Casimiroperator. We summarize the latest results on the universal enveloping algebras ofLie algebras and Lie triple systems. In particular, we study the Casimir element asthe element in the center of the universal enveloping algebras of Lie triple systems.The main purpose of the study is to give a new characterization of semi-simple Lietriple systems among the quadratic Lie triple systems, by using Casimir elements.Finally, we study the relationship between Lie triple systems and Jordan triple sys tems.