### Background

I would like to know if there is some slick machinery to solve the following representation-theoretic problem.

Let $\left(V,\langle-,-\rangle\right)$ be a finite-dimensional real inner product space and let $\mathfrak{so}(V)$ denote the Lie algebra of skewsymmetric endomorphisms.

Let $\mathfrak{g}$ be a Lie algebra and $\rho: \mathfrak{g} \to \mathfrak{so}(V)$ be a Lie algebra homomorphism. This makes $V$ into an orthogonal $\mathfrak{g}$-module.

Let $\sigma: \mathfrak{so}(V) \to \operatorname{End}(\Delta)$ denote an irreducible spinor representation of $\mathfrak{so}(V)$. Then the composition $\sigma \circ \rho : \mathfrak{g} \to \operatorname{End}(\Delta)$ of the representations turns $\Delta$ into a $\mathfrak{g}$-module.

### Question

Is there a nice description of this module? And in particular about its decomposition into irreducibles?

### Contextualisation

This sort of problem arises often in my line of work; although usually with $\mathfrak{g}$ itself being an orthogonal Lie algebra and $V$ itself an irreducible spinor representation. In other words, I end up considering *spinors of spinors* and this is useful in determining the structure of representations ("supermultiplets") of Lie superalgebras appearing in supersymmetric field theories. I usually work this out "by hand" (or using LiE) in terms of roots and weights, but was wondering whether there was some more conceptual machinery available.