Let
$g\,\mapsto \,{{g}^{*}}$
denote an involution on a group
$G$
. For any (commutative, associative) ring
$R$
(with 1),
$*$
extends linearly to an involution of the group ring
$RG$
. An element
$\alpha \,\in \,RG$
is symmetric if
${{\alpha }^{*}}\,=\,\alpha $
and skew-symmetric if
${{\alpha }^{*}}\,=\,-\alpha $
. The skew-symmetric elements are closed under the Lie bracket,
$[\alpha ,\,\beta ]\,=\,\alpha \beta \,-\,\beta \alpha $
. In this paper, we investigate when this set is also closed under the ring product in
$RG$
. The symmetric elements are closed under the Jordan product,
$\alpha \,o\,\beta \,=\,\alpha \beta \,+\beta \alpha $
. Here, we determine when this product is trivial. These two problems are analogues of problems about the skew-symmetric and symmetric elements in group rings that have received a lot of attention.