  In this paper, we study a family of infinite-dimensional Lie algebras
$\widehat{X}_{S}$, where $X$ stands for the type: $A,B,C,D$, and $S$ is an
abelian group, which generalize the $A,B,C,D$ series of trigonometric Lie
algebras. Among the main results, we identify $\widehat{X}_{S}$ with what are
called the covariant algebras of the affine Lie algebra
$\widehat{\mathcal{L}_{S}}$ with respect to some automorphism groups, where
$\mathcal{L}_{S}$ is an explicitly defined associative algebra viewed as a Lie
algebra. We then show that restricted $\widehat{X}_{S}$-modules of level $\ell$
naturally correspond to equivariant quasi modules for affine vertex algebras
related to $\mathcal{L}_{S}$. Furthermore, for any finite cyclic group $S$, we
completely determine the structures of these four families of Lie algebras,
showing that they are essentially affine Kac-Moody Lie algebras of certain
types.
