Multiplying two generalised octonions is a slow process by hand, but it can be done. I just did it, in fact, but it took me a long time. We'll be multiplying the two octonions:
A = (a + bi0 + ci1 + di2 + ei3 + fi4 + gi5 + hi)
B = (j + k0 + l1 + m2 + n3 + p4 + q5 + r6)
To begin, we shall multiply out the terms in full, being careful not to change the order of the units as octonions, like quaternions, are not commutative.
AB = (a + b0 + c1 + d2 + e3 + f4 + g5 + h)(j + k0 + l1 + m2 + n3 + p4 + q5 + r6)
= aj + aki0 + ali1 + ami2 + ani3 + api4 + aqi5 + ari6
+ bji0 + bki12 + bli0i1 + bmi0i2 + bni0i3 + bpi0i4 + bqi0i5 + bri0i6
+ cji1 + cki1i0 + cli12 + cmi1i2 + cni1i3 + cpi1i4 + cqi1i5 + dri1i6
+ dji2 + dki2i0 + dli2i1 + dmi22 + dni2i3 + dpi2i4 + dqi2i5 + dri2i6
+ eji3 + eki3i0 + eli3i1 + emi3i2 + eni32 + epi3i4 + eqi3i5 + eri3i6
+ fji4 + fki4i0 + fli4i1 + fmi4i2 + fni4i3 + fpi42 + fqi4i5 + fri4i6
+ gji5 + gki5i0 + gli5i1 + gmi5i2 + gni5i3 + gpi5i4 + gqi52 + gri5i6
+ hji6 + hki6i0 + hli6i1 + hmi6i2 + hni6i3 + hpi6i4 + hqi6i5 + hri62
Now we can apply the triplet rules detailed under octonion to multiply out the units, obtaining:
= aj + aki0 + ali1 + ami2 + ani3 + api4 + aqi5 + ari6
+ bji0 - bk + bli3 + bmi6 - bni1 + bpi5 - bqi4 - bri2
+ cji1 - cki3 - cl + cmi4 + cni0 - cpi2 + cqi6 - cri5
+ dji2 - dki6 - dli4 - dm + dni5 + dpi1 - dqi3 + dri0
+ eji3 + eki1 - eli0 - emi5 - en + epi6 + eqi2 - eri3
+ fji4 - fki5 + fli2 - fmi1 - fni6 - fp + fqi0 + fri3
+ gji5 + gki4 - gli6 + gmi3 - gni2 - gpi0 - gq + gri1
+ hji6 + hki2 + hli5 - hmi0 + hni4 - hpi3 - hqi1 - hr
Finally, we can group things together by their i coeffecient, to finish with:
= aj - bk - cl - dm - en - fp - gq - hr
i0(ak + bj + cn + dr - el + fq - gp - hm)
i1(al - bn + cj + dp + ek - fm + gr - hq)
i2(am - br - cp + dj + eq + fl - gn + hk)
i3(an + bl - ck - dq + ej + fr + gm - hp)
i4(ap - bq + cm - dl - er + fj + gk + hn)
i5(aq + bp - cr + dn - em - fk + gj + hl)
i6(ar + bm + cq - dk + ep - fn - gl + hj)
And there you have it. Any two Cayley numbers, multiplied together.