This is perhaps a simple tensor calculus problem -- but I just can't see why...
I have notes (in GR) that contains a proof of the statement
In space of constant sectional curvature, $K$ is independent of position.
Here
$$R_{abcd}\equiv K(x)(g_{bd}g_{ac}-g_{ad}g_{bc})$$ where $R_{abcd}$ is the Riemann curvature tensor and $g_{ab}$ is the metric of the spacetime.
The proof goes like this:
Contract the defining equation with $g^{ac}$, giving $$R_{bd}=3Kg_{bd}.$$ and so on.
Problem is I don't understand why the contraction gives $$R_{bd}=3Kg_{bd}.$$ I can see the first term gives $$g^{ac}g_{bd}g_{ac}=4g_{bd}$$ since it's 4D spacetime. But as far as I can tell, the second term gives $g^{ac}g_{ad}g_{bc}=\delta_{bd}$ which is not necessarily $g_{bd}$.
Where have I gone wrong?