# Classification of involutions of the lattice $H\oplus H(k)^{\oplus2}$ for $k=5,6$?

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 $k=5,6$.

-