$\DeclareMathOperator\RRe{Re}\DeclareMathOperator\Spin{Spin}\DeclareMathOperator\Sym{Sym}$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)](https://mathoverflow.net/questions/243889/a-representation-of-spin9-1)). One of them is the image of $j$.



Quadratic forms on $\mathbb{R}^{16}$ coming from octonions
I'm revising my answer because whether the image of the map $j$ that the OP defines is equal to the $10$-dimensional subspace $H$ of $\mathrm{Sym}^2(\mathbb{R}^{16})$ that is invariant under $\mathrm{Spin}(9,1)$ depends on which particular action of $\mathrm{Spin}(9,1)$ on $\mathbb{R}^{16}$ one chooses.


First, independent of the map $j$ there is a characterization of $H$ in terms of the $\mathrm{Spin}(9,1)$-action on $\mathbb{R}^{16}=\mathbb{O}^2$ in other terms that may be what the OP wants. This goes as follows: $\mathrm{Spin}(9,1)$ preserves an $8$-sphere $\Sigma$ of $8$-dimensional subspaces $L\subset\mathbb{O}^2$ such that $\mathbb{O}^2\setminus\{0\}$ is foliated smoothly by the punctured planes $L\setminus\{0\}$ for $L\in\Sigma$. Each of these $8$-planes $L$ carries a 'natural' definite quadratic form defined up to a positive multiple such that, for each $g\in \mathrm{Spin}(9,1)$ and $L\in\Sigma$, the induced isomorphism $g:L\to g(L)$ identifies the two quadratic forms up to a multiple.


Then $H$ consists of the quadratic forms on $\mathbb{O}^2$ that restrict to each $L\in\Sigma$ to be a multiple of its 'natural' quadratic form.


All of this follows from the description of $\mathrm{Spin}(9,1)\subset\mathrm{SL}(\mathbb{R}^{16})=\mathrm{SL}(\mathbb{O}^{2})$ given at the end of my [Notes on spinors in low dimension](https://arxiv.org/abs/2011.05568), where the Lie algebra of $\mathrm{Spin}(9,1)$ is described as
$$
{\frak{spin}}(9,1) = 
\left\{\pmatrix{
 a\_1 +x\,I\_8 & C\,R\_{\bf w} \cr
 C\,L\_{\bf x} & a\_3 -x\,I\_8 }\ 
 :\ 
\matrix{ x\in\mathbb{R},\cr {\bf w},{\bf x}\in\mathbb{O},\cr
 \ a\in{\frak{spin}}(8)}\ \right\}
\subset{\frak{sl}}(16,\mathbb{R})\,,
$$
where $C$ denotes conjugations in the octonions, $R\_{\bf w}$ denotes right multiplication by ${\bf w}$ in the octonions, $L\_{\bf x}$ denotes left multiplication by ${\bf x}$ in the octonions, and the assignments $a\mapsto a\_k$ for $k=1,3$ are Lie algebra isomorphisms $\mathfrak{spin}(8)\to \mathfrak{so}(8)$ described in the Notes referenced above.


For example, the subspaces $L\in\Sigma\simeq S^8 = \mathbb{OP}^1$ are of the form either $L\_\infty = \{\ (q,0)\ | \ q\in\mathbb{O\ }\}$ or of the form $L\_w = \{\ (wq, \bar q)\ |\ q\in\mathbb{O}\ \}$ for some $w\in\mathbb{O}$. Alternatively, they can be described as either of the form $L'\_\infty = \{\ (0,q)\ | \ q\in\mathbb{O\ }\}$ or of the form $L'\_x = \{\ (\bar q, qx)\ |\ q\in\mathbb{O}\ \}$ for some $x\in\mathbb{O}$. (Note that $L\_w = L'\_x$ when $w\bar x = 1$ while $L\_0 = L'\_\infty$ and $L\_\infty = L'\_0$.) [Here, I am writing elements of $\mathbb{O}^2$ as pairs $(p,q)$ to save space, but they really should be written in the form $\textstyle\begin{pmatrix}p\\ q\end{pmatrix}$ to be consistent with the matrix notation used above for ${\frak{spin}}(9,1)$.]


Now, with this identification of $\mathbb{R}^{16}$ with $\mathbb{O}^2$, $H$ is not equal to the image of $j$, as is easy to check.


On the other hand, if we take the conjugate of $\mathrm{Spin}(9,1)$ under the matrix
$$
\pmatrix{I\_8 & 0 \cr 0 & C }
$$
(which does not belong to $\mathrm{Spin}(9,1)$), then, for the $H$ associated to this conjugate subgroup, the image of $j$ *is* equal to $H$.


