Page 3.
By the way, the index doubling and index cycling properties of the previous page
are octonion automorphisms (elements of the automorphism group G_{2}; in fact
they give rise to the full 21 element subgroup of G_{2} that maps octonion units to units).
The D(+) multiplication table is based on the cyclic identity
e_{a} e_{a+1} = e_{a2} = e_{a+5},
and the C(+) table on the identity
e_{a} e_{a+1} = e_{a4} = e_{a+3}.
In both cases e_{0} represents the octonion identity (so be careful of
the modulo 7 index property, as 7 modulo 7 = 7 in these multiplication rules).
So on the D(+) side, using index doubling twice,
e_{1} e_{2} = e_{6} implies
e_{2} e_{4} = e_{5},
e_{4} e_{1} = e_{3}
(one more application of index doubling gets you back where you started);
and using
index cycling,
e_{1} e_{2} = e_{6} implies
e_{2} e_{3} = e_{7},
e_{3} e_{4} = e_{1},
e_{4} e_{5} = e_{2},
e_{5} e_{6} = e_{3},
e_{6} e_{7} = e_{4},
e_{7} e_{1} = e_{5}.
Using index doubling on the rules above, we arrive at three useful cyclic
identities for each table:
D(+)

C(+)

e_{a} e_{a+1} = e_{a+5}

e_{a} e_{a+1} = e_{a+3}

e_{a} e_{a+2} = e_{a+3}

e_{a} e_{a+2} = e_{a+6}

e_{a} e_{a+4} = e_{a+6}

e_{a} e_{a+4} = e_{a+5}

Given index doubling only the first of each of these pairs of three rules
is necessary, but in any case, with these rules and the index doubling and
cycling automorphisms in hand, determining a product using either of these
tables is very easy. In general, e_{a} e_{b} will give a positive result
if ba is a power of 2 (1, 2, or 4) modulo 7, and it will give a negative result
if ba is minus a power of 2 (6, 5 or 3) modulo 7.
And that about does it. All you have to watch out for is the modulo 7 stuff.
For example, using index doubling, given
e_{1} e_{7} = e_{3}
on the C(+) side, we get
e_{2} e_{7} = e_{6}.
That is, 2x7=7 modulo 7.
One other note here: the unit element
( 1 + e_{1} + e_{2} + ... + e_{7})/8^{1/2}
and its conjugate have special roles to play in octonion mathematics.
Note that they are index doubling and cycling invariant.
Page 2.

Page 4.

Contents

Octoshop III page.