Determined the remaining sign $\pm 1$ in that prop:

$L_{e_7} L_{e_6} L_{e_5} L_{e_4} L_{e_3} L_{e_2} L_{e_1} = + 1$ ]]>added statement and Clifford-theoretic proof (here) that the consecutive left product by all the seven imaginary generators acts as the identity

]]>stated the Clifford action of the imaginary octonions induced by left multiplication (here)

]]>made explicit the definition of real and imaginary octonions here

]]>added statement and proof (here) that the octonions are alternative

]]>Added a statement (here) concerning projecting out $\mathbb{H}$ from $\mathbb{O}$.

]]>added statement and proof (here) that the product of all the seven imaginary quaternions with each other is $\pm 1$.

]]>added the actual definition to *octonions*

Added the definition of “basic triples” of octonions, and the statement that they form a torsor over $Aut(\mathbb{O}) = G_2$.

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