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What is the restriction of the (complex) spin representation of $so(n+m)$ to the block diagonal subalgebra $so(n)\times so(m)$?

A naive guess is that it is the (complex) tensor product of the two spin representations. This has the right dimension when n and m aren't both odd.

When they are both odd, then this has half the dimension of the (dirac) spin representation, so maybe this tensor product is equal to one of the (weyl) half-spin representations, but which one?

And what is the restriction of the other weyl representation?

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