  We show that the splitting feature of the Einstein tensor, as the first term
of the Lovelock tensor, into two parts, namely the Ricci tensor and the term
proportional to the curvature scalar, with the trace relation between them is a
common feature of any other homogeneous terms in the Lovelock tensor. Motivated
by the principle of general invariance, we find that this property can be
generalized, with the aid of a generalized trace operator which we define, for
any inhomogeneous Euler-Lagrange expression that can be spanned linearly in
terms of homogeneous tensors. Then, through an application of this generalized
trace operator, we demonstrate that the Lovelock tensor analogizes the
mathematical form of the Einstein tensor, hence, it represents a generalized
Einstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnet
gravity as an another version of string-inspired gravity.
