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On that Google Nature paper, I'm not sure if the 5x5 matrix improvement is the more significant result, or that they figured out a way to optimize directly to SIMD/AVX instructions, resulting in faster execution even if not necessarily fewer operations

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The result is not interesting for ordinary numbers, where the classical ("slower") method is faster. When each element of the 5x5 matrix is itself a matrix, it could be important.

So AVX ops for the method are not interesting.

No... using divide and conquer matrix multiplication with this gives n^2.83. Strassens algorithm is n^2.79. This is good fir 5 by 5 matrices... which are not terribly interesting.

It uses many more additions. When the elements are themselves matrices, additions are a lot cheaper than multiplications, so that is OK. But with ordinary numbers, additions cost about the same as multiplications, so trading off a few multiplications for a lot more additions loses.

I thought it was when you were multiplying two matrices where the elements are 5x5 rather than a 5x5 matrix containing embedded matrices of arbitrary length, no?

The technique works for either

But usefully for only one.

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