The special case of 2-Scorza varieties are sometimes called Severi varieties, after Francesco Severi.
Zak showed that k-Scorza varieties are the projective varieties of the rank 1 matrices of rank k simple Jordan algebras.
The Severi varieties are the non-singular varieties of dimension n (even) in PN that can be isomorphically projected to a hyperplane and satisfy N=3n/2+2.
These 4 Severi varieties can be constructed in a uniform way, as orbits of groups acting on the complexifications of the 3 by 3 hermitian matrices over the four real (possibly non-associative) division algebras of dimensions 2k = 1, 2, 4, 8.
Zak proved that the only Severi varieties are the 4 listed above, of dimensions 2, 4, 8, 16.