A projective representation
of a variety of the first row of the Freudenthal–Tits magic square
can be obtained as the absolute geometry of a (symplectic) polarity
of the projective
representation
of a variety one cell below. In this paper, we extend this geometric connection between
and
by showing that any nondegenerate quadric
of maximal Witt index
containing
gives rise to a
variety isomorphic to
, in the
sense that the symplecta of
contained in totally isotropic subspaces of
are the absolute symplecta of a unique (symplectic) polarity
of
.
Except for the smallest case, we also show that any nondegenerate quadric containing
has
maximal Witt index; and in the largest case, we obtain that there are only
three kinds of possibly degenerate quadrics containing the Cartan variety
.
