Let σ1 and σ2 be commuting involutions of a semisimple algebraic group G. This yields a \documentclass[12pt]{minimal}
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\begin{document}$$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$\end{document}-grading of \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g} $$\end{document} = Lie(G), \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g}={\oplus_{i,j=0,1 }}{{\mathfrak{g}}_{ij }} $$\end{document}, and we study invariant-theoretic aspects of this decomposition. Let \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$\end{document} be the \documentclass[12pt]{minimal}
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\begin{document}$$ {{\mathbb{Z}}_2} $$\end{document}-contraction of \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g} $$\end{document} determined by σ1. Then both σ2 and σ3 := σ1σ2 remain involutions of the non-reductive Lie algebra \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g}\left\langle {{\sigma_1}} \right\rangle $$\end{document}. The isotropy representations related to \documentclass[12pt]{minimal}
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\begin{document}$$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_2}} \right) $$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$ \left( {\mathfrak{g}\left\langle {{\sigma_1}} \right\rangle, {\sigma_3}} \right) $$\end{document} are degenerations of the isotropy representations related to \documentclass[12pt]{minimal}
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\begin{document}$$ \left( {\mathfrak{g},{\sigma_2}} \right) $$\end{document} and \documentclass[12pt]{minimal}
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\begin{document}$$ \left( {\mathfrak{g},{\sigma_3}} \right) $$\end{document}, respectively. These degenerated isotropy representations retain many good properties. For instance, they always have a generic stabiliser and their algebras of invariants are often polynomial. We also develop some theory on Cartan subspaces for various \documentclass[12pt]{minimal}
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\begin{document}$$ {{\mathbb{Z}}_2} $$\end{document}-gradings associated with the \documentclass[12pt]{minimal}
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\begin{document}$$ {{\mathbb{Z}}_2}\times {{\mathbb{Z}}_2} $$\end{document}-grading of \documentclass[12pt]{minimal}
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\begin{document}$$ \mathfrak{g} $$\end{document} and study the special case in which σ1 and σ2 are conjugate.
