  Let $\sigma_1$ and $\sigma_2$ be commuting involutions of a semisimple
algebraic group $G$. This yields a $Z_2\times Z_2$-grading of $\g=\Lie(G)$,
$\g=\bigoplus_{i,j=0,1}\g_{ij}$, and we study invariant-theoretic aspects of
this decomposition. Let $\g<\sigma_1>$ be the $Z_2$-contraction of $\g$
determined by $\sigma_1$. Then both $\sigma_2$ and $\sigma_3:=\sigma_1\sigma_2$
remain involutions of the non-reductive Lie algebra $\g<\sigma_1>$. The
isotropy representations related to $(\g<\sigma_1>, \sigma_2)$ and
$(\g<\sigma_1>, \sigma_3)$ are degenerations of the isotropy representations
related to $(\g, {\sigma_2})$ and $(\g, {\sigma_3})$, respectively. We show
that 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 $Z_2$-gradings associated with the $Z_2\times
Z_2$-grading of $\g$.
