by Markus Rost (Notes, April 2003, 7 pages)
By a theorem of Albert a central simple algebra of degree four contains a biquadratic subalgebra. The major point is to prove the existence of a quadratic subalgebra.
This can be shown based on the fact from projective geometry that, given four generic lines in three-space, there are exactly two lines which meet them all.
In these notes we describe a related algebraic construction.
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