Classical orthogonal decomposition of the special linear Lie algebra

In this research, we consider a classical orthogonal decomposition of the special linear Lie algebra sln  over a finite field Fq . For n = 2,3,4, we search for a necessary and sufficient condition for which sln(Fq) has a classical orthogonal decomposition and construct the decomposition explicitly. Up to isomorphism, we also find the number of classical orthogonal decomposition for sln(Fq)  (if exists) and find the maximum number of classical Cartan subalgebra components if the decomposition does not exist. Next, we find the complete description of Fq  for which sln(Fq) has a classical orthogonal decomposition. Finally, we generalize ideas to the case of  sln  over other general commutative rings such as principal idea rings.