Classical orthogonal decomposition of a modular ${\mathfrak{s}\mathfrak{l}}_n$

An orthogonal decomposition problem of Lie algebras over the complex numbers has been studied since the 1980s. It has many applications and relations to other areas of mathematics and sciences. In this paper, we consider this decomposition problem over a field of prime characteristic. We define a classical orthogonal decomposition of a modular Lie algebra and construct it for ${\mathfrak{s}\mathfrak{l}}_n$ under certain sufficient conditions. Additionally, we provide more detailed analysis of the problem when $n=2$ and $3$.