  We derive an explicit formula for the well-known Chern-Moser-Weyl tensor for
nondegenerate real hypersurfaces in complex space in terms of their defining
functions. The formula is considerably simplified when applying to
"pluriharmonic perturbations" of the sphere or to a Fefferman approximate
solution to the complex Monge-Amp\`ere equation. As an application, we show
that the CR invariant one-form $X_{\alpha}$ constructed recently by Case and
Gover is nontrivial on each real ellipsoid of revolution in $\mathbb{C}^3$,
unless it is equivalent to the sphere. This resolves affirmatively a question
posed by these two authors in 2017 regarding the (non-) local CR invariance of
the $\mathcal{I}'$-pseudohermitian invariant in dimension five and provides a
counterexample to a recent conjecture by Hirachi.
