  This paper investigates the toral stabilizer of a finite-dimensional
restricted Lie algebra $(\mathfrak{g},[p])$, defined over an algebraically
closed field $k$ of positive characteristic. We compute the group of toral
stabilizers of some simple Lie algebras and give a new characterization of the
solvable restricted Lie algebras. Moreover, we determine the subalgebra
structure of exceptional simple Lie algebras in good characteristic. We also
provide applications, which concern the distribution of weight spaces.
