Let $M$ be an even lattice of signature $ (b^{+},b^{-})$ , with dual $M'$ . We use $v$ to denote an isometry from $M \otimes\textbf{R}$ to $\textbf{R}^{b^+,b^-}$ . We write $v^+$ and $v^-=v^{+\perp}$ for the inverse images of $\textbf{R}^{b^+,0}$ , $\textbf{R}^{0,b^-}$ under $v$ . The projection of $\lambda \in M \otimes\textbf{R}$ into a subspace $v^{\pm}$ is denoted by $\lambda_{v^{\pm}}$ , so that $\lambda=\lambda_{v^+}+\lambda_{v^-}$ . Does someone know what $\textbf{R}^{b^+,b^-}$ means and how to find the inverse images of it under $v$ .
Thanks for the help .