Extending Lasenby's embedding of octonions in space‐time algebra Cl(1,3), to all three‐ and four dimensional Clifford geometric algebras Cl(p,q), n=p+q=3,4
Eckhard Hitzer
Abstract
We study the embedding of octonions in the Clifford geometric algebra for space‐time STA , as suggested by Anthony Lasenby at AGACSE 2021. As far as possible, we extend the approach to similar octonion embeddings for all three‐ and four‐dimensional Clifford geometric algebras , . Noticeably, the lack of a quaternionic subalgebra in seems to prevent the construction of an octonion embedding in this case and necessitates a special approach in . As examples, we present for the nonassociativity of the octonionic product in terms of multivector grade parts with cyclic symmetry and show how octonion products and involutions can be combined to make the opposite transition from octonions to the Clifford geometric algebra and how octonionic multiplication can be represented with (complex) biquaternions or Pauli matrix algebra.