Hi everyone,

I was doing some lengthy computation involving covariant derivatives and gamma matrices acting on spinors.

And I often encounter expressions containing the commutator of covariant derivatives. as following

$ R*{a b c d} \gamma*{a b} \nabla*{c}{\nabla*{d}{\epsilon}} $

where because of the anti-symmetry on the indices $c$ and $d$ of the Riemann tensor.

The summation on theses give rise to the commutator of covariant derivatives acting on spinors and I want to use the identity (or definition) such as

$ [\nabla*{c}, {\nabla*{d}] {\epsilon}} = \frac{1}{4} R*{c d e f} \gamma*{e f} \epsilon$

what should i do? of course, I can use the substitute() to convert the whole term like

substitute(_, $ R*{a b c d} \gamma*{a b} \nabla*{c}{\nabla*{d}{\epsilon}} -> \frac{1}{4} R*{a b c d} R*{c d e f} \gamma_{e f} \epsilon $);

But this is not clever at all and I need to identify manually term by term. Which is obviously impossible in slightly more complicated examples.

Looking forward to learn and to read your suggestions. Thank you if you can give me some hint.