  We present the subalgebra structure of sl(3,O), a particular real form of e6
chosen for its relevance to particle physics and its close relation to
generalized Lorentz groups. We use an explicit representation of the Lie group
SL(3,O) to construct the multiplication table of the corresponding Lie algebra
sl(3,O). Both the multiplication table and the group are then utilized to find
various nested chains of subalgebras of sl(3,O), in which the corresponding
Cartan subalgebras are also nested where possible. Because our construction
involves the Lie group, we simultaneously obtain an explicit representation of
the corresponding nested chains of subgroups of SL(3,O).
