Comment by gjm11

My point about #5 was just that their emphasis on "complex numbers" isn't necessarily (and I think probably isn't) an admission that their algorithm can't be applied recursively.

If it can be, then it is a genuine advance in the sense that it yields faster large-n matmuls than were previously available just by recursive application of a 4x4 matmul algorithm.

None of which, to be clear, makes it OK that they don't make any mention of earlier work that does 4x4 matmuls over commutative rings faster than their algorithm does. I suspect they didn't check the literature carefully enough before publication. In any case, good scholarship means being clear about what in your work is a genuine advance on what's already known and what isn't, and they failed to do that here.