  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 $\mathfrak{sl}_n$ under certain sufficient conditions. Additionally, we
provide more detailed analysis of the problem when $n = 2$ and $3$.
