The highest dimension of commutative subalgebras in Chevalley algebras

Let $L_\Phi(K)$ denotes a Chevalley algebra with the root system $\Phi$ over a field $K$. In 1945 A. I. Mal'cev investigated the problem of describing abelian subgroups of highest dimension in complex simple Lie groups. He solved this problem by transition to complex Lie algebras and by reduction to the problem of describing commutative subalgebras of highest dimension in the niltriangular subalgebra. Later these methods were modified and applied for the problem of describing large abelian subgroups in finite Chevalley groups. The main result of this article allows to calculate the highest dimension of commutative subalgebras in a Chevalley algebra $L_\Phi (K)$ over an arbitrary field.