A note on orthogonal decomposition of sln over commutative rings

The orthogonal decomposition (OD) of special linear Lie algebras over the complex numbers was
first explored in the early 1980s and has gained renewed interest in the past decade due to its applications in
quantum information theory. In this study, we investigate the OD of special linear Lie algebras over commutative rings with identity. Additionally, we present examples of specific rings that facilitate the existence
of such ODs for these Lie algebras.