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We can identify the Lorentz algebra as two su(2)'s. Hence from QM I'm convinced that the representation of the Lorentz algebra can be of dimension (2s_1 + 1)(2s_2 + 1).

The Weyl spinor is two dimensional so it's either a (s_1, s_2) = (1/2, 0) or a (0,1/2) representation (i.e. left or right handed).

But it then seems (since one representation of the su(2)s is the trivial) as if I only need to specify three parameters when lorentz transforming my Weyl spinors.. What happened to my choice of three rotations and three boosts?

Thanks! :)