You make an important point, but I understood the parent as, "what is the smallest integer expressible as the sum of two cubes of Cayley-Dickson algebras of or

You make an important point, but I understood the parent as, "what is the smallest integer expressible as the sum of two cubes of Cayley-Dickson algebras of order n?" So you're allowed to use complex/quaternions etc. in the cubes, but the number has to come from those well-ordered integers.