Let $K$ be a field with $2\in K^\times$, let $A$ be a separable $K$-algebra (i.e. $A$ is finite-dimensional, semisimple and its center is an etale $K$-algebra), and let $\sigma:A\to A$ be a $K$-involution.

Consider $\mathbf{U}(A,\sigma)$, the unitary algebraic $K$-group of $(A,\sigma)$; it is the affine algebraic $K$-group with sections given by $$\mathbf{U}(A,\sigma)(L)=U(A_L,\sigma_L):=\{a\in A_L\,:\,a^\sigma a=1\},$$ where $A_L=A\otimes_K L$, $\sigma_L=\sigma\otimes_K\mathrm{id}_L$. Denote by $\mathbf{U}^0(A,\sigma)$ the neutral connected component of $\mathbf{U}(A,\sigma)$.

Let $\overline{K}$ be an algebraic closure of $K$. Using the fact that $A_{\overline{K}}$ is a product of matrix algebras over $\overline{K}$, it is not difficult to see that:

$\mathbf{U}(A,\sigma)$ is a form a prodcut of copies of $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{O}_k$ ($m,n,k$ can vary).

$\mathbf{U}^0(A,\sigma)$ is a form a prodcut of copies of $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{SO}_k$ ($m,n,k$ can vary).

A "folklore" fact asserts that the converse also holds. That is, for all $m_1,\dots,m_r,n_1,\dots,n_s,k_1,\dots,k_t\in \mathbb{N}$, any $K$-form of $\mathbf{GL}_{m_1}\times\dots\times \mathbf{GL}_{m_r}\times \mathbf{Sp}_{2n_1}\times\dots\times\mathbf{Sp}_{2n_s}\times\mathbf{O}_{k_1}\times\dots\times \mathbf{O}_{k_t}$ is of the form $\mathbf{U}(A,\sigma)$, with $(A,\sigma)$ uniquely determined up to isomorphism, and similarly with $\mathbf{U}^0(A,\sigma)$ when one replaces $\mathbf{O}$ with $\mathbf{SO}$.

**My first question** is whether there is an explicit reference for this statement in the literature?
Notice that it should be possible to deduce this statement from section 26 in the *Book of Involutions*, say, by passing to the simply-connected covering. This will presumably require some work to bridge the difference between $\mathbf{SL}_m$ and $\mathbf{GL}_m$, and also to eliminate the tritalitarian forms of $\mathbf{Spin}_8$. I am asking for a reference which will require less adjustments.

**My second question** is whether the scheme version of the "fact" above is known in the literature?
In more detail, we can replace $K$ with a scheme $S$ (with $2$ invertible on $S$) and assume that $A$ is a locally-free separable $\mathcal{O}_S$-algebra.
(This is same as saying that there are $t\geq 0$ and $n_1,\dots,n_t\in\Gamma(S,\mathbb{N})$ such that $A$ and $\prod_{i=1}^t\mathrm{Mat}_{n_i\times n_i}(\mathcal{O}_S)$ are locally isomorphic relative to the etale topology.)
Is it true that any (etale) form of a product of copies of the group $S$-schemes $\mathbf{GL}_m$, $\mathbf{Sp}_{2n}$, $\mathbf{O}_k$ (where $m,n,k\in\Gamma(S,\mathbb{N})$ can vary) is of the form $\mathbf{U}(A,\sigma)\to \mathrm{Spec}S$?
I would also be happy for a proof in case a reference cannot be found.