For an even integral lattice $L$ (of arbitrary signature, non-degenerate but not necessarily unimodular), consider roots (vectors of square $2$) and their orthogonal hyperplanes. First, I'd like to know if the collection of root hyperplanes is always locally finite. (I'm not actually sure what this means, but I have come across this condition in many papers. At the very least, it should mean that every non-zero point in $L \otimes \mathbb{R}$
 has a neighborhood meeting a finite number of the hyperplanes.) Second, if the above is true, I can take the complement of all of these hyperplanes to be left with a bunch of connected components and I'd like to know if choosing one of them always gives a fundamental domain for the group generated by reflections in the root hyperplanes. I know that this is the case for finite and affine root systems, and am wondering if this is a general fact.