$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$Let $\mathcal{H}_2(\mathbb{O})$ denote the (10-dimensional) real vector space of octonionic Hermitian matrices of size 2. Recall that a matrix $(a_{ij})$ with octonionic entries is called Hermitian if $a_{ji}=\bar a_{ij}$.

One has a linear imbedding $j\colon \mathcal{H}_2(\mathbb{O})\to \Sym^2(\mathbb{R}^{16})$ to the space of real quadratic forms on $\mathbb{R}^{16}=\mathbb{O}^2$ given by $$(j(A))(\xi)=\sum_{i,j=1}^2\RRe((\bar\xi_iA_{ij})\xi_j),$$ where $\xi=(\xi_1,\xi_2)\in \mathbb{O}^2$.

**QUESTION. I am looking for a characterization of the image of $j$.**

Ideally I need a description analogous to the complex case as follows. The space $\mathcal{H}_n(\mathbb{C})$ of complex Hermitial matrices is imbedded into the space of real quadratic forms on $\mathbb{R}^{2n}=\mathbb{C}^n$ via the similar map $$j'(A)(\xi)=\sum_{i,j=1}^n \bar\xi_iA_{ij}\xi_j.$$ It is known that its image consists precisely of real quadratic forms invariant under the multiplication $\xi\mapsto z\cdot\xi$ for any complex number with $|z|=1$. However, as far as I can see, this description does not seem to generalize to the octonionic situation.

**REMARK.** I am aware of a representation theoretical characterization of the image of $j$. There is an action of the group $\Spin(1,9)$ on $\mathbb{R}^{16}=\mathbb{O}^2$ (see e.g. p.29 here https://arxiv.org/pdf/math/0105155v4.pdf ). The symmetric square representation in $\Sym^2(\mathbb{R}^{16})$ is a sum of exactly two non-isomorphic irreducible subspaces (see the answer to this post A representation of Spin(9,1)). One of them is the image of $j$.