  Attempts to extend our previous work using the octonions to describe
fundamental particles lead naturally to the consideration of a particular real,
noncompact form of the exceptional Lie group E6, and of its subgroups. We are
therefore led to a description of E6 in terms of 3x3 octonionic matrices,
generalizing previous results in the 2x2 case. Our treatment naturally includes
a description of several important subgroups of E6, notably G2, F4, and (the
double cover of) SO(9,1), An interpretation of the actions of these groups on
the squares of 3-component "Cayley spinors" is suggested.
