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The Harke paper
http://www.harke.org/ps/intro.ps.gz
is directed to more profound
problems than are normally faced in undergraduate mechanics,
So,IMO, foisting this stuff on helpless undergraduates (and their
teachers) qualifies as corruption of youth in the worst sense of ....
the positive
subalgebra (to be defined) of the Clifford Algebra of V(4,(4)) is just the
ordinary vector algebra that you have all used since childhood.