  The First and Second Representation Theorem for sign-indefinite quadratic
forms are extended. We include new cases of unbounded forms associated with
operators that do not necessarily have a spectral gap around zero. The kernel
of the associated operators is determined for special cases. This extends
results by Grubi\v{s}i\'c, Kostrykin, Makarov and Veseli\'c in [Mathematika 59
(2013), 169--189].
