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I created this octonion calculator to facilitate my continuing casual investigation of the Leech lattice.
In particular, when using the multiplication table employed above, the element
F = (1 - e₁ - e₂ - ... - e₇)/2
(the element b above is twice this) can be used to generate from a particular pair of octonions in the
8-d laminated lattice E₈ a third octonion such the the trio lies in the 24-d laminated lattice (Leech).
Note: in my paper Octonions: Invariant Leech Lattice Exposed (which can be found on the
internet somewhere) I used the conjugate of F to this end. But I was using an octonion multiplication
table in that paper dual to that employed above, and the third element generator (F) had to be changed
to accomodate this difference. Note that F and it's dual are examples of elements invariant under index
cycling and doubling. The 32 input fields defining the octonions a,b,c,d can be anything you want them
to be. The result at right will be the components of a(b(cd)) or (ab)(cd) (order of parentheses matters!).
By the way, using the d given above, the trio (d/2, d/2, (bd)/4) is an element of my representation of
the Leech lattice (since a = c = 1 when the program opens, bd = a(b(cd)) at the start (see element at right)).