  Cubic sevenfolds are examples of Fano manifolds of Calabi-Yau type. We study
them in relation with the Cartan cubic, the $E_6$-invariant cubic in
$\PP^{26}$. We show that a generic cubic sevenfold $X$ can be described as a
linear section of the Cartan cubic, in finitely many ways. To each such "Cartan
representation" we associate a rank nine vector bundle on $X$ with very special
cohomological properties. In particular it allows to define auto-equivalences
of the non-commutative Calabi-Yau threefold associated to $X$ by Kuznetsov.
Finally we show that the generic eight dimensional section of the Cartan cubic
is rational.
