  The Goldman Lie algebra of an oriented surface was defined by Goldman. By the
natural involution that opposes the orientation of curves, the Goldman Lie
algebra becomes a $\mathbb{Z}_{2}$-graded Lie algebra. Its even part is
isomorphic to the Thurston-Wolpert-Goldman Lie algebra or, briefly, the TWG Lie
algebra. Chas and Kabiraj proved the center of the TWG Lie algebra is generated
by the class of the unoriented trivial loop and the classes of unoriented loops
parallel to boundary components or punctures. The center of the even part can
be rephrased as the set of elements of the even part annihilated by all the
elements of the even part. We also prove some similar statements for the
remaining 3 cases involving the odd part. Moreover, we compute the elements of
the symmetric algebra and the universal enveloping algebra of the Goldman Lie
algebra annihilated by all the even elements of the Goldman Lie algebra, and
those annilated by all the odd elements.
