  A 4-dimensional Riemannian manifold M, equipped with an additional tensor
structure S, whose fourth power is minus identity, is considered. The structure
S has a skew-circulant matrix with respect to some basis of the tangent space
at a point on M. Moreover, S acts as an isometry with respect to the metric g.
A fundamental tensor is defined on such a manifold (M,g,S) by g and by the
covariant derivative of S. This tensor satisfies a characteristic identity
which is invariant to the usual conformal transformation. Some curvature
properties of (M,g,S) are obtained. A Lie group as a manifold of the considered
type is constructed. A Hermitian manifold associated with (M,g,S) is also
considered. It turns out that it is a locally conformal K\"{a}hler manifold.
