Let
$$\mathfrak{g} = W_1 $$
be the Witt algebra over an algebraically closed field k of characteristic p > 3; and let
be the commuting variety of g. In contrast with the case of classical Lie algebras of P. Levy [J. Algebra, 2002, 250: 473–484], we show that the variety
is reducible, and not equidimensional. Irreducible components of
and their dimensions are precisely given. As a consequence, the variety
is not normal.
