I am using usual physics notations and I guess the physics motivations of this question are obvious.

Let a basis of the $SO(n,m)$ Lie algebra be denoted by $S^{\mu \nu}$ and the Lie algebra be, $[S^{\mu \nu},S^{\lambda \rho}] = i(g^{\mu \rho}S^{\nu \lambda} + g^{\nu \lambda}S^{\mu \rho} - g^{\mu \lambda}S^{\nu \rho} - g^{\nu \rho}S^{\mu \lambda})$ where $g$ is the matrix $diag(-1,-1,..m-times..,-1,1,1,..n-times..,1)$.

At least in the $m=1$ (Lorentz) case it is true that if one can find a set of matrices $\Gamma^\mu$ such that they satisfy the "corresponding" Clifford algebra, $[\Gamma^\mu, \Gamma ^\nu] = 2g^{\mu \nu}$ then a representation of the $SO(n,1)$ Lie algebra is given by $S^{\mu \nu} = \frac{i}{4}[\Gamma ^\mu , \Gamma ^\nu]$

Does the above construction work for arbitrary $m$ , especially $m = 0,2$ and what is the global understanding for why this should work?

The representation $S^{\mu \nu} = \frac{i}{4}[\Gamma ^\mu , \Gamma ^\nu]$ is how the $SO(n,1)$ Lie algebra acts on its spinorial representations. (..those representations whose weights are given by a $[\frac{n+1}{2}]$ tuple of $\pm \frac{1}{2}$..)

What is the $m \neq 1$ generalization of the above? (...one case that I have often seen used are these two earlier questions of mine..)

Now my main question is to understand how the above construction - at least for the most familiar case of $n=3, m=1$ - gives an alternative to using the language of tensors.

Like to give probably the most used example - if $F_{\mu \nu}$ is a $4$-dimensional antisymmetric rank $2$ tensor then one defines the quantities $F_{\alpha \beta}$ and $\bar{F} _ {\dot{\alpha} \dot{\beta}}$ s.t $F_{\alpha \beta} = (S^{\mu \nu}) _ {\alpha \beta} F_{\mu \nu}$ and $\bar{F} _ {\dot{\alpha} \dot{\beta}} = (S^{\mu \nu})_{\dot{\alpha} \dot{\beta}} F_{\mu \nu}$

What exactly is the group/representation theoretic meaning behind defining these $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$?

In what sense is knowing the $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ equivalent to knowing the $F_{\mu \nu}$?

In what sense is $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ the self-dual ($\frac{1}{2}(F + *F)$) and the anti-self-dual ($\frac{1}{2}(F - *F)$) parts of the tensor $F$?

I guess from here one can also explain why the self-dual part is thought to be in the $(1,0)$ representation and the anti-self-dual part is thought to be in the $(0,1)$ representation of either the $SL(2,\mathbb{C}) \times SL(2,\mathbb{C})$ (..isometry group of the complexified Minkowski space..) or of $SU(2) \times SU(2)$ (..in in Euclidean four dimensional space..)

Is there an analogue of the $F_{\alpha \beta}$ and $\bar{F}_{\dot{\alpha} \dot{\beta}}$ for general $SO(n,m)$ and arbitary dimension higher rank tensors ?

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