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We show that the splitting feature of the Einstein tensor, as the first termof the Lovelock tensor, into two parts, namely the Ricci tensor and the termproportional to the curvature scalar, with the trace relation between them is acommon feature of any other homogeneous terms in the Lovelock tensor. Motivatedby the principle of general invariance, we find that this property can begeneralized, with the aid of a generalized trace operator which we define, forany inhomogeneous Euler-Lagrange expression that can be spanned linearly interms of homogeneous tensors. Then, through an application of this generalizedtrace operator, we demonstrate that the Lovelock tensor analogizes themathematical form of the Einstein tensor, hence, it represents a generalizedEinstein tensor. Finally, we apply this technique to the scalar Gauss-Bonnetgravity as an another version of string-inspired gravity.Comment: 16 pages, added refs., corrected a mistyped eq. (the last one in  p12), TeX file (using phyzzx.tex

Lovelock tensor as generalized Einstein tensor