One can take as basis for the eight-dimensional real vector space the scalar 1, three anti-commuting vectors , three bivectors , and the pseudo-scalar . The algebra multiplication is associative and unitary (there exists a multiplicative unit, 1). The pseudo-scalar M squares to . Scalar and pseudo-scalar commute with everything. The three basis vectors , by definition, square to . The three basis bivectors square to . Take any unit bivector v. It satisfies hence . If the space could be given a norm such that the norm of a product is the product of the norms, we would have hence either or (or both), implying that either or (or both), implying that or , neither of which are true.
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