Moreno Variant of Cayley-Dickson Construction of Octonions

Using the variant Cayley-Dickson formula (a,b)(c,d) = (ac - d*b, da + bc*) gives as special cases:

(e_i, 0)(0, 1) = (0, e_i) and (0, 1)(e_i, 0) = (0, e_i*)
(0, 1)(0, e_i) = (-e_i*, 0) and (0, e_i)(0, 1) = (-e_i, 0)
(e_i, 0)(0, e_j) = (0, e_je_i) and (0, e_j)(e_i, 0) = (0, e_je_i*)
(0, e_i)(0, e_j) = (-e_j*e_i, 0)

Applying these to the right-handed quaternions extends their multiplication table thus (where the "canonical" mapping is assumed,
i.e. e_i ↔ (e_i, 0) for i = 1,2,3; e₄ ↔ (0, 1); e_i+4 ↔ (0, e_i)):

	e₀	e₁	e₂	e₃
e₀	e₀	e₁	e₂	e₃
e₁	e₁	-e₀	e₃	-e₂
e₂	e₂	-e₃	-e₀	e₁
e₃	e₃	e₂	-e₁	-e₀

→

	e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇
e₀	e₀	e₁	e₂	e₃	e₄	e₅	e₆	e₇
e₁	e₁	-e₀	e₃	-e₂	e₅	-e₄	-e₇	e₆
e₂	e₂	-e₃	-e₀	e₁	e₆	e₇	-e₄	-e₅
e₃	e₃	e₂	-e₁	-e₀	e₇	-e₆	e₅	-e₄
e₄	e₄	-e₅	-e₆	-e₇	-e₀	e₁	e₂	e₃
e₅	e₅	e₄	-e₇	e₆	-e₁	-e₀	-e₃	e₂
e₆	e₆	e₇	e₄	-e₅	-e₂	e₃	-e₀	-e₁
e₇	e₇	-e₆	e₅	e₄	-e₃	-e₂	e₁	-e₀

with triads:
(1,2,3),
(1,4,5),
(1,7,6),
(2,4,6),
(2,5,7),
(3,4,7),
(3,6,5)

giving signmask bits (lowest order bit for 1,2,3 etc.): 1000100 = 0x44