Four-dimensional tensor identities of low order for the Weyl and Ricci tensors

It is shown that the new tensor identity recently discovered by Bonanos, and some other tensor identities recently investigated, are consequences of a very simple and mathematically trivial (but subtle) identity highlighted some years ago by Lovelock. Lovelock's identity gives a tensor identity of first order in Weyl-like tensors, and a tensor identity of second order in Ricci-like tensors, from which higher order identities, such as those recently studied, can easily be constructed.