Abstract: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
E_6
, and of its subgroups. We are therefore led to a description of
E_6
in terms of
3\times 3
octonionic matrices, generalizing previous results in the
2\times 2
case. Our treatment naturally includes a description of several important subgroups of
E_6
, notably
G_2
,
F_4
, 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.
Keywords: octonions,
E_6
, exceptional Lie groups, Dirac equation
AMS Subject Classification: 17C90 17A35 22E70