Regular Cartan subalgebras and nilpotent elements in restricted Lie algebras

Let $\mathfrak G$ be a finite-dimensional restricted Lie algebra over an algebraically closed field of characteristic $p>0$. It is proved that any two Cartan subalgebras with maximal toral part in $\mathfrak G$ can be obtained from each other by means of a finite chain of elementary transformations that are similar in form to the exponents of the inner root derivations of $\mathfrak G$. The following theorem plays an important role in the proof:
Theorem. {\it Let $s$ be a toral rank of $\mathfrak G$ and $e_1,\dots,e_n$ a basis of $\mathfrak G$. There exists $\nu\in\mathbf Z_+$ and homogeneous polynomials $f_0,\dots,f_{s-1},$ in $n$ variables$,$ such that
$$ x^{[p^{s+\nu}]}=\sum_{i=0}^{s-1}f_i(x_1,\dots,x_n)x^{[p^{i+\nu}]} $$
$($here $x=x_1e_1+\dots+x_ne_n$ and $\deg f_i=p^{s+\nu}-p^{i+\nu}).$}
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