The commuting variety of pglₙ

We are considering the commuting variety of the Lie algebra pglₙ over an algebraically closed field of characteristic $p >0$, namely the set of pairs \ (A,B) ∈ pglₙ × pglₙ [A,B]=0 \. We prove that if $n=pr$, then there are precisely two irreducible components, of dimensions n²+r-1 and n²+n-2.