Simple algebras with involution, and unitary groups

Let $A$ be a central simple algebra on which an involutory antiautomorphism $S$ is given whose restriction to the center $K$ of $A$ is not the identity. Let $\Sigma(A^*)$ be the subgroup of the multiplicative group $A^*$ of $A$ generated by the elements $x\in A^*$ such that $x^S=x$, let $Nrd_{A/K}\colon A\to K$ be the reduced norm mapping of $A$ into $K$, and let $\Sigma'(A^*)$ be the subgroup of $A^*$ generated by the elements $x\in A^*$ whose reduced norm is invariant with respect to $S$. This paper considers the problem of when the groups $\Sigma'(A^*)$ and $\Sigma(A^*)$ coincide.
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