algebraic cone; monoid action; group action; orbit; action with unique fixed point; algebra automorphism; stable ideal; irreducible algebraic variety semifree action; algebraic realization; compact Lie group; orthogonal representation; real algebraic \(G\) variety; smooth \(G\) manifold; algebraically realized; equivariant diffeomorphism; \(G\)-vector bundles; product of a group of odd order and a 2-torus; equivariant map; equivariantly homotopic; equivariant bordism class 