  Let $\mathfrak{g}$ be a simple Lie algebra defined over an algebraically
closed field $k$ of characteristic $p$. Fix an integer $r>1$ and suppose that
$V_1,\ldots,V_r$ are irreducible closed subvarieties of $\mathfrak{g}$. Let
$C(V_1,\ldots,V_r)$ be the closed variety of all the pairwise commuting
elements in $V_1\times\cdots\times V_r$. This paper studies the dimension and
irreducibility of such varieties with various $V_i$ in a Lie algebra
$\mathfrak{g}$. In particular, we complete the problem for the case when
$V_i$'s are either $\overline{\mathcal{O}_{\text{sub}}}$ the closure of the
subregular orbit or $\mathcal{N}$ the nilpotent cone of any rank two Lie
algebra $\mathfrak{g}$. A result on the dimension of these mixed commuting
varieties is generalized for higher ranks. Finally, we apply our calculations
to study properties of support varieties for a simple module over the $r$-th
Frobenius kernels of $G$.
