Denote by $\mathbb{C}_{\mathrm{sp}}$ be the split complex numbers. This is isomorphic to the direct sum $\mathbb{R}\oplus\mathbb{R}$ with norm $N(a,b)=ab$ and conjugation $\overline{(a,b)}=(b,a)$. Denote by $\mathbb{H}_{\mathrm{sp}}$ the split quaternions. This is isomorphic to the matrix algebra $M_2(\mathbb{R})$ with norm $N(X)=\det X$ and conjugation given by the formula $\overline{X}=JX^TJ^{-1}$ where $J$ is either $90^{\circ}$ rotation matrix.
On either $\mathbb{C}_{\mathrm{sp}}^{p,q}$ or $\mathbb{H}_{\mathrm{sp}}^{p,q}$ there is a sesquilinear form given by
$$ \langle x,y\rangle=(\overline{x_1}y_1+\cdots+\overline{x_p}y_p)-(\overline{x_{p+1}}y_{p+1}+\cdots+\overline{x_{p+q}}y_{p+q}). $$
We will abbreviate $\mathbb{C}_{\mathrm{sp}}^{n,0}$ and $\mathbb{H}_{\mathrm{sp}}^{n,0}$ by removing $0$ from the superscript.
Writing $n=p+q$, I have been able to determine the following isomorphisms:
$$ \mathrm{SU}(\mathbb{C}_{\mathrm{sp}}^ {p,q})\cong\mathrm{SL}(\mathbb{R}^n), \quad \mathrm{U}(\mathbb{C}_{\mathrm{sp}}^{p,q})\cong\mathrm{GL}(\mathbb{R}^n), \quad \mathrm{U}(\mathbb{H}_{\mathrm{sp}}^n)\cong\mathrm{Sp}(\mathbb{R}^{2n}).$$
And was similarly able to determine $\mathrm{SL}$ and $\mathrm{GL}$ over $\mathbb{C}_{\mathrm{sp}}$ and $\mathbb{H}_{\mathrm{sp}}$. Moreover, it turns out that there should be an isomorphism $\mathrm{U}(\mathbb{H}_{\mathrm{sp}}^2)\cong\mathrm{U}(\mathbb{H}_{\mathrm{sp}}^{1,1})$ since they are both $\mathrm{Spin}(3,2)$.
However, I'm unsure if there are other $\mathrm{U}(\mathbb{H}_{\mathrm{sp}}^{p,q})$ of indefinite signature $(p,q)$ that are expressible using classical matrix Lie groups. Are there?