  We utilize recently introduced linear programming bounds for the energy of
periodic configurations in $\mathbb{R}^d$ to construct configurations which are
universally optimal among those of the form $\omega_4+L_\beta$, where
$\omega_4$ is a 4-point multiset and $L_\beta\subseteq \mathbb{R}^2$ is a
rectangular lattice. For two values of the parameter $\beta$, the
configurations are obtained from the $A_2$ lattice. Other values of $\beta$
yield the third, and largest cardinality, class of examples of
$\Lambda$-universally optimal configurations which do not arise from lattices
$\Lambda$ known or conjectured to be universally optimal in $\mathbb{R}^d$.
