$5$ dimensional space over $\mathbb{R}$

When coming up with a double cover of $SO(5)$, I used conjugation by matrices of the form $$\begin{pmatrix} r & q\\ \overline{q} & r \end{pmatrix}$$ where $r\in\mathbb{R}$ and $q$ is a quaternion. These matrices are clearly $5$ dimensional over $\mathbb{R}$, but I'm wondering if someone can identify this space by name so that I can find more information.

Edit: I should have also added that for any matrix $A$ of this form, $A=A^*$ where $*$ denotes conjugate transpose. However, this requirement follows from how $A$ was defined, so mentioning it again doesn't add information, but rather is a (potentially) useful observation.

I feel compelled to write this answer, based on the fact that the answer which was accepted makes no sense.

What the OP mentions seems to be an embedding of $\mathbb{R}^5$ inside the space $M$ of hermitian $2\times 2$ matrices over the quaternions, which is $Sp(2)$-equivariant: where $Sp(2)$ acts on $\mathbb{R}^5$ via the canonical double cover $Sp(2) \to SO(5)$ and acts on $M$ by the conjugation.

This is analogous to the following well-known construction.

You can exhibit the double cover $SL(2,\mathbb{C}) \to SO(3,1)_0$ as follows. You identify $\mathbb{R}^4$ with the real vector space $H$ of hermitian $2\times 2$ matrices. (Minus) the determinant of a hermitian $2 \times 2$ matrix defines a quadratic form on $H$ coming from an inner product of signature $(3,1)$. Define an action of $SL(2,\mathbb{C})$ on $H$ by $h \mapsto s h s^\dagger$, for $h \in H$ and $s \in SL(2,\mathbb{C})$. This is a linear transformation of $H$ which preserves the determinant (since $s$ has unit determinant) and hence defines an element of $O(3,1)$. Since $SL(2,\mathbb{C})$ is connected, it actually defines a surjective map $SL(2,\mathbb{C}) \to SO(3,1)_0$ to the identity component. The kernel of this map is the group of order $2$ generated by $-I$.

The analogy breaks down in that there is no quaternionic determinant in general, but perhaps for matrices in $M$ it can be defined. I have not checked.

There are similar double covers which can be explicitly described in this way: e.g., $SL(2,\mathbb{R}) \to SO(2,1)_0$ and $SL(2,\mathbb{H}) \to SO(1,5)_0$.