Spinor representations decomposed under subgroups in Joe's big book

The question is about the spinor representation decomposed under subgroups. It's a common technique in string theory when parts of dimensions are compactified and ignored, and we are only interested in the remaining sub-symmetry. I'm learning it from the appendix B in Polchinski's big book volume II. For a particular decomposition, SO(2k+1,1) → SO(2l+1,1) × SO(2k-2l) (B.1.43), the Weyl spinors decompose as the formula (B.1.44), 2^k → (2^l, 2^k-l-1)+(2'^l,2'^k-l-1) and 2'^k → (2'^l, 2^k-l-1)+(2^l,2'^k-l-1), where 2^k and 2'^k are the Weyl representations of Lorentz group SO(2k+1,1) with chirality +1 and -1 respectively.



Specifically on the case SO(9,1)→ SO(5,1)× SO(4) with decomposetions 16 → (4,2)+(4',2'), which appears at (B.6.3). My question is **the contradiction with minimum representations**. By checking Majorana and Weyl conditions, the minimum spinors for d=6 and d=4 have 8 and 4 components respectively. (Ref. table B.1 Polchinski) So How can you find (4,2) representation for SO(5,1)× SO(4)?



Also, I'm very interested in the prove of (B.1.44) under (B.1.43)? How is it proved by comparing the eigenvalues of $\Gamma^{+}\Gamma^{-}-\frac{1}{2}$ as claimed by Polchinski?

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