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I want to derive the Weyl tensor along the lines of this derivation, but I am unable to complete it. (I am only interested in $4$ dimension for now.)

Every contraction I perform gives either $0=R + 3 \lambda L$ or $0=0$ but not an additional equation to calculate $L$ and $\lambda$.

Does anyone know a source where I can find this derivation in detail or point me to the crucial point I am missing?

Thank you!

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First, drop $\lambda$, because it is useless (can be absorbed into $L$). Second, try contracting in $a$ and $c$ only. This contraction should also be zero.

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Ok, then I obviously get $0=R + 3 L$, which i can write as $0=R_{bd}g^{bd} + 3 L_{bd}g^{bd}$, but how to continue now? –  Michal May 17 '13 at 21:55
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Don't do double contractions. You can get more information by just doing the single contraction $C_{abcd}g^{ac}=0$. –  Edward Hughes May 17 '13 at 22:10
    
You mean, that I should stop at $0=R_{bd} + \frac{1}{2}\bigl(2 L_{bd} + g_{bd}L\bigr)$ and not contract then with $g^{bd}$? –  Michal May 18 '13 at 10:04
    
@Michal, yes. If you now recall your $R+3L=0$, you should be able to find $L_{bd}$. –  Peter Kravchuk May 18 '13 at 19:42
    
Thanks, I should have seen that! –  Michal May 19 '13 at 9:04
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