  We study the orthogonality of the generalized eigenspaces of an
Ornstein--Uhlenbeck operator $\mathcal L$ in $\mathbb{R}^N$, with drift given
by a real matrix $B$ whose eigenvalues have negative real parts. If $B$ has
only one eigenvalue, we prove that any two distinct generalized eigenspaces of
$\mathcal L$ are orthogonal with respect to the invariant Gaussian measure.
Then we show by means of two examples that if $B$ admits distinct eigenvalues,
the generalized eigenspaces of $\mathcal L$ may or may not be orthogonal.
