Standard Cayley-Dickson Construction of Octonions

Using the traditional Cayley-Dickson formula (a,b)(c,d) = (ac - db*, a*d + cb) 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_i*e_j) and (0, e_j)(e_i, 0) = (0, e_ie_j)
(0, e_i)(0, e_j) = (-e_je_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),
(1,5,4),
(1,7,6),
(2,6,4),
(2,5,7),
(3,7,4),
(3,6,5)

giving signmask bits (lowest order bit for 1,2,3 etc.): 1101110 = 0x6e