  The normal symmetric triality algebras (STA's) and the normal Lie related
triple algebras (LRTA's) have been recently introduced by the second author, in
connection with the principle of triality. It turns out that the unital normal
LRTA's are precisely the structurable algebras extensively studied by Allison.
  It will be shown that the normal STA's (respectively LRTA's) are the algebras
that coordinatize those Lie algebras whose automorphism group contains a copy
of the alternating (resp. symmetric) group of degree 4.
