  Different ways of representing the group $SU(3)$ within a Geometric Algebra
approach are explored. As part of this we consider characteristic multivectors
for $SU(3)$, and how these are linked with decomposition of generators into
commuting bivectors. The setting for this work is within a 6d Euclidean
Clifford Algebra. We then go on to consider whether the fundamental forces of
particle physics might arise from symmetry considerations in just the 4d
geometric algebra of spacetime -- the STA. As part of this, a representation of
$SU(3)$ is found wholly within the STA, involving preservation of a bivector
norm. We also show how Octonions can be fully represented within the Spacetime
Algebra, which we believe will be useful in making them understandable and
accessible to a new community in Physics and Engineering. The two strands of
the paper are drawn together in showing how preserving the octonion norm is the
same as preserving the timelike part of the Dirac current of a particle. This
suggests a new model for the symmetries preserved in particle physics.
Following on from work by G\"unaydin and G\"ursey on the link between quarks,
and octonions, and by Furey on chains of octonionic multiplications, we show
how both of these fit well within our scheme, and give some wholly STA versions
of the operations involved, which in the cases considered have easily
understandable equivalents in terms of 4d geometry. Links with larger groups
containing $SU(3)$, such as $G_2$ and $SU(8)$, are also considered.
