  We introduce a natural notion of determinant in matrix JB$^*$-algebras, i.e.,
for hermitian matrices of biquaternions and for hermitian $3\times 3$ matrices
of complex octonions. We establish several properties of these determinants
which are useful to understand the structure of the Cartan factor of type $6$.
As a tool we provide an explicit description of minimal projections in the
Cartan factor of type $6$ and a variety of its automorphisms.
