Let's try to build from scratch the SUSY commutator $[Q_\alpha^I, P_\mu]$. We know that the result of this commutator must be a fermonic generator belonging to $(1/2, 0)\otimes(1/2,1/2) \simeq (1, 1/2)\oplus (0, 1/2)$, so that I am allowed to write \begin{equation} [Q_\alpha^i, P_\mu] = A_{\alpha \mu}^i + B_{\alpha \mu}^i, \end{equation} with a priori two arbitrary sets of fermionic generators in the relevant irreps of $SL(2, \mathbb{C})$. Since there are no $(1, 1/2)$ sermonic operators I can set $A=0$, so that \begin{equation} [Q_\alpha^i, P_\mu] = B_{\alpha \mu}^i. \end{equation} Since for each choice of indices $B_{\alpha \mu}^i$ must be in $(0, 1/2)$, I can write it as a linear combination of the $\bar{Q}^{\dot{\alpha} j}$s, thus \begin{equation} [Q_\alpha^i, P_\mu] = b^{ij}_{\alpha \dot{\alpha}\mu}\bar{Q}^{\dot{\alpha} j}. \end{equation} Now, for each choice of $i,j$ and $\mu$, we have a 2x2 complex matrix $b^{ij}_{\alpha \dot{\alpha}\mu}$, which we can then expand on the basis of such matrices given by the Pauli matrices, hence \begin{equation} [Q_\alpha^i, P_\mu] = b^{ij}_{\mu \nu}(\sigma^\nu)_{\alpha \dot{\alpha}}\bar{Q}^{\dot{\alpha} j}. \end{equation} It is not so obvious to me that $b$ must be proportional to the identity in the $\mu \nu$ indices as is assumed in every discussion of such commutators (e.g. Constructing SUSY algebra via index structure).
Nevertheless, Lorentz transforming both sides of the equation and using the mathematical identity $D_{(1/2,0)}(\Lambda)^{-1}\sigma ^\mu D_{(0, 1/2)}(\Lambda) = \Lambda^{\mu}_{\; \nu}\sigma^\nu $, consistency requires that $[\Lambda, b^{ij}]_{\mu \nu}=0$ for all $\Lambda$, thus $b^{ij}_{\mu \nu}=b^{ij}\delta_{\mu\nu}$, and \begin{equation} [Q_\alpha^i, P_\mu] = b^{ij}(\sigma_\mu)_{\alpha \dot{\alpha}}\bar{Q}^{\dot{\alpha} j} \end{equation} as expected (not by me but by everyone else, apparently).
My question is then:
is there a way to rigorously prove that the index structure one would naively guess is the right one without having to go through the burden of checking consistency every single time?
I'm interested in this since for more complicated (anti)commutators I cannot find a way to reproduce the natural index structure as I managed above. For instance the equation (2.72) of Bertolini's "Lectures on Supersymmetry" (https://people.sissa.it/~bertmat/susycourse.pdf), \begin{equation} \{Q_\alpha ^i, Q_\beta^j\}=\epsilon_{\alpha \beta} Z^{ij} + \epsilon_{\beta \gamma}(\sigma^{\mu \nu})_\alpha ^{\; \gamma} J_{\mu \nu} Y^{ij} \end{equation} is almost completely mysterious to me.
