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Let $H$ denote the hyperbolic lattice (rank 2 lattice generated by $e,f$ such that $e^2=f^2=e.f-2=0$). Let $k >0$ be an integer. Is it possible to classify involutions $\iota$ of the lattice $$ L:=H\oplus H(k)^{\oplus2} $$ under the condition that rank$L^\iota=2$ and sign$(L^\iota)=(1,1)$?

I am particularly interested in the cases when $k=5,6$.

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