Summary: Let \(\mathbf k\) be an algebraically closed field of characteristic 0 and \(A\) be a finitely generated associative \(\mathbf k\)-algebra, in general noncommutative. One assigns to \(A\) a sequence of commutative \(\mathbf k\)-algebras \(\mathcal{O}(A,d), d=1,2,3,\ldots\), where \(\mathcal{O}(A,d)\) is the coordinate ring of the space \(\mathrm{Rep}(A,d)\) of \(d\)-dimensional representations of the algebra \(A\). A double Poisson bracket on \(A\) in the sense of Van den Bergh [Trans. Am. Math. Soc. 360, 5711–5799 (2008)] is a bilinear map \(\{\!\{-,-\}\!\}\) from \(A\times A\) to \(A^{\otimes 2}\), subject to certain conditions. Van den Bergh showed that any such bracket \(\{\!\{-,-\}\!\}\) induces Poisson structures on all algebras \(\mathcal{O}(A,d)\). We propose an analog of Van den Bergh’s construction, which produces Poisson structures on the coordinate rings of certain subspaces of the representation spaces \(\mathrm{Rep}(A,d)\). We call these subspaces the involutive representation spaces. They arise by imposing an additional symmetry condition on \(\mathrm{Rep}(A,d)\) – just as the classical groups from the series B, C, D are obtained from the general linear groups (series A) as fixed point sets of involutive automorphisms.