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In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of supersymmetric boundary conditions in N=4 Super Yang-Mills in four dimensions in performed. One of the main points in this analysis is the breaking of the R-symmetry at the boundary from $SO(6)$ to $SO(3)\times SO(3)$, which is explained on page 6.

My main interest is the set of operators given in equation (2.7). It is claimed that the action of $W=SO(1,2)\times SO(3)\times SO(3)$ commutes with the operators $$ B_0=\Gamma_{456789}\\B_1=\Gamma_{3456}\\B_2=\Gamma_{3789}, $$ (where the subscripts on right-hand-sides indicate completely antisymmetrized products of 10D gamma matrices. Note that $SO(1,2)$ acts on the indices $012$, and the two $SO(3)$ groups act on $456$ and $789$ respectively.)

How does one prove that this is true?

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The $B_0$ is the chirality matrix of $SO(6)\rightarrow SO(3)_X\times SO(3)_Y$, namely $\Gamma_{456789}$.

The $B_1$ is the product between the gamma matrices of $SO(3)_{X}$, namely $\Gamma_{456}$, with $\Gamma_3$.

The $B_{2}$ is the product between the gamma matrices of $SO(3)_{Y}$, namely $\Gamma_{789}$, with $\Gamma_3$

You need to prove that $\Gamma_{456789}$, $\Gamma_{3456}$ and $\Gamma_{3789}$ commutes with $\Gamma_{\mu\nu}$ for $\mu$ and $\nu$ equal $0,1,2$ xor $4,5,6$ xor $7,8,9$. This is the same thing as showing that the number of indices that are equal between $\Gamma_{\mu\nu}$ and $\Gamma_{456789}$ is always even. That the number of indices that are equal between $\Gamma_{\mu\nu}$ and $\Gamma_{3456}$ are always even. That the number of indices that are equal between $\Gamma_{\mu\nu}$ and $\Gamma_{3789}$ are always even.

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