Hawking's and Ellis' derivation of the form of Einstein's field equations

Question

On pages 72-73 of the book "The large scale structure of space-time" Hawking and Ellis show while determining the form of the field equations of general relativity that there is a relation of the form $R_{ab} = K_{ab}$, where $K_{ab}$ is a tensorial function of the energy-momentum tensor $T_{ab}$ and the metric $g_{ab}$.

They then state that $K_{ab}$ is of the form $$K_{ab} = \kappa \left(T_{ab}-\frac{1}{2} Tg_{ab} \right) + \Lambda g_{ab}$$ for some constants $\kappa$ und $\Lambda$. Here they use apart from the stated tensorial property the following facts.

1. The contracted Bianchi identities ${K_{a}^{\,b}}_{;b} = \frac{1}{2} K_{;b}$ hold.
2. The only first order identities satisfied by the energy-momentum tensor are the conservation equations ${T_{a}^{\,b}}_{;b} = 0$.
3. Perhaps from Postulate (b) in the book the symmetry of the energy-momentum tensor $T_{ab}$.
4. Maybe also some regularity assumptions on the function $K_{ab}$ are implicitly assumed.

We have tried unsuccessfully to give a mathematically rigorous proof of this fact. In the vacuum case, i.e. the energy-momentum tensor vanishes and $K_{ab}$ therefore only depends on the metric, one can argue as follows:

Writing $g = (g_{ab})$ and $K = (K_{ab})$ the tensorial property translates into the identity $K(SgS^T) = SK(g)S^T$ for all regular matrices $S$. It follows from Sylvester's law and the tensorial property that it suffices to determine the value of $$A = K\left(\begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}\right) \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.$$ Now, from the tensorial property one obtains that $A$ commutes with every element of the Lorentz group and therefore must be a multiple of the identity, for example by Schur's lemma. Therefore one has $$K(g) = K\left(S \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} S^T \right) = S A \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} S^T = \Lambda S \begin{pmatrix} -1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 &1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix} S^T = \Lambda g.$$ So in this special case the assertion follows only from the tensorial property.

However, we do not know how to deal with the general case. Can anyone give us a reference or a hint? Moreover, does one need property 3)?