  The commuting variety of a reductive Lie algebra $\mathfrak{g}$ is the
underlying variety of a well defined subscheme of
$\mathfrak{g}\times\mathfrak{g}$. In this note, it is proved that this scheme
is normal and Cohen-Macaulay. In particular, its ideal of definition is a prime
ideal. As a matter of fact, this theorem results from a so called Property (P)
for a simple Lie algebra. This property says that some cohomology complexes are
exact.
