  We adapt for algebraically closed fields $k$ of characteristic greater than
$2$ two results of Voisin, on the decomposition of the diagonal of a smooth
cubic hypersurface $X$ of dimension $3$ over $\mathbb C$, namely: the
equivalence between Chow-theoretic and cohomological decompositions of the
diagonal of those hypersurfaces and the fact that the algebraicity (with
$\mathbb Z_2$-coefficients) of the minimal class $\theta^4/4!$ of the
intermediate jacobian $J(X)$ of $X$ implies the Chow-theoretic decomposition of
the diagonal of $X$. Using the second result, the Tate conjecture for divisors
on surfaces defined over finite fields predicts, via a theorem of Schoen, that
every smooth cubic hypersurface of dimension $3$ over the algebraic closure of
a finite field of characteristic $>2$ admits a Chow-theoretic decomposition of
the diagonal.
