  Every symmetric generating functional of a convolution semigroup of states on
a locally compact quantum group is shown to admit a dense unital $*$-subalgebra
with core-like properties in its domain. On the other hand we prove that every
normalised, symmetric, hermitian conditionally positive functional on a dense
$*$-subalgebra of the unitisation of the universal C$^*$-algebra of a locally
compact quantum group, satisfying certain technical conditions, extends in a
canonical way to a generating functional. Some consequences of these results
are outlined, notably those related to constructing cocycles out of convolution
semigroups.
