In the paper Supersymmetric Boundary Conditions in N=4 Super Yang-Mills Theory by Gaiotto and Witten, an in-depth analysis of boundary conditions in N=4 Super Yang-Mills in four dimensions in performed.
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?