This question arises when I tried to understand Chapter 2 of the celebrated book "Smooth Compactification of Locally Symmetric Varieties" by Ash--Mumford--Rapoport--Tai.

The setting is as follows: consider a finite dimensional real vector space $V$ and a symmetric cone $C$ in $V$ (symmetric=open,self-dual and homogeneous). Then they proceed to consider the automorphism group $G$ of $(V,C)$, namely the group of linear automorphisms of $V$ that preserve $C$.

It is fairly easy to show that $G$ is real reductive, in the sense that $G\subseteq GL(V)$ is closed in the Euclidean topology and is stable under conjugate transpose.

But when they considered the boundary components of $C$, they implicitly use that $G$ is an algebraic group (or more precisely, there is an algebraic group $\mathcal{G}$ over $\mathbb{R}$ such that $G=\mathcal{G}(\mathbb{R})^+$). For example, this is used on Page 54 to guarantee the representability of $Norm(C_0)$.
I cannot really figure out a proof of the algebraicity of $G$. By some easy manipulation, we can readily reduce the problem to simple symmetric cones. So my question is: is the automorphism group of a simple symmetric cone an algebraic group?

In the classical case, this seems easy and follows from the explicit computations in Faraut--Korányi's book. But what about the semi-classical case and the exceptional case?