  The triplet contribution is computed to the 1 and 2 $^1S^\text{e}_0$ states
of the He atom, to the $1\ ^1S^\text{e}_0$ state of the Li$^+$ and Be${^{2+}}$
ions, and to the $X\ ^1\Sigma_\text{g}^+$ ground state of the H$_2$ molecule by
extensive use of double-group symmetry (equivalent to $LS$ coupling for the
atomic systems) during the course of the variational solution of the no-pair
Dirac-Coulomb-Breit wave equation. The no-pair Dirac-Coulomb-Breit energies are
converged within a sub-parts-per-billion relative precision using an explicitly
correlated Gaussian basis optimized to the non-relativistic energies. The
$\alpha$ fine-structure constant dependence of the triplet sector contribution
to the variational energy is $\alpha^4E_\text{h}$ at leading order, in
agreement with the formal perturbation theory result available from the
literature.
