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split quaternion in nLab

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Definition

The split-quaternions are an algebra over the real numbers. Every split quaternion qq may be represented as

q=a 0+a 1i+a 2j+a 3k q = a_0 + a_1 i + a_2 j + a_3k

where the basis elements satisfy the following products:

×\times	i	j	k
i	-1	k	-j
j	-k	+1	-i
k	j	i	+1

and conjugation t *=a 0−a 1i−a 2j−a 3kt^* = a_0 - a_1 i - a_2 j - a_3k.

These are closely related to the quaternions, as the generators satisfy similarly-looking relations. They are obtained from the split-complex numbers through the generalization of the Cayley-Dickson construction.