A representation of Spin(9,1)

Question

Let $Spin(9,1)$ denote the universal (double) cover of $SO(9,1)$. $Spin(9,1)$ acts linearly on $\mathbb{R}^{16}$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ).

Consider the induced action of $Spin(9,1)$ in the space of symmetric quadratic forms on $\mathbb{R}^{16}$, i.e. in $Sym^2((\mathbb{R}^{16})^*)$.

How to decompose (the complexification of) this space into irreducible components? In particular, how many irreducible components there exist?

Remark. In the answer to this post Decomposition into irreducible components of a representation of $Spin(9)$ it was stated that the decomposition of the above space under the action of the smaller group $Spin(9)\subset Spin(9,1)$ contains exactly 3 irreducible components which are pairwise non-isomorphic.

Answer 1

In this case if your 16-dimensional $\mathrm{Spin}(9,1)$-representation is the one of highest weight $(0,0,0,0,1)$, then the $\mathrm{Spin}(9,1)$-irreducible decomposition of its symmetric square is just into two pieces: The 10-dimensional piece $\mathbb{R}^{9,1}$ isomorphic to the standard (Lorentzian) representation of $\mathrm{SO}(9,1)$ (with highest weight $(1,0,0,0,0)$) and the 126-dimensional representation of highest weight $(0,0,0,0,2)$. This latter representation can be shown to be isomorphic to the space of self-dual $5$-forms on $\mathbb{R}^{9,1}$, i.e., $\Lambda^5_+(\mathbb{R}^{9,1})\subset \Lambda^5(\mathbb{R}^{9,1})$.