  We continue the study undertaken in \cite{DV} of the exceptional Jordan
algebra $J = J_3^8$ as (part of) the finite-dimensional quantum algebra in an
almost classical space-time approach to particle physics. Along with reviewing
known properties of $J$ and of the associated exceptional Lie groups we argue
that the symmetry of the model can be deduced from the Borel-de Siebenthal
theory of maximal connected subgroups of simple compact Lie groups.
