code class$=$"asciimath"$>$Help me check the Jacobi identity regards the Lie bracket $\backslash$sum$\_($cyclic$)[$X$,[$Y$,$Z$]]=0.$ This notation means that one permutes X$,$Y$,$Z cyclically and one takes the sum of the resulting terms. Written out, $\backslash$sum$\_($cyclic$)[$X$,[$Y$,$Z$]]=[$X$,[$Y$,$Z$]]+[$Y$,[$Z$,$X$]]+[$Z$,[$X$,$Y$]].$ This is the exercise $14\mathrm{.}11$ from An Introduction to Manifolds by Tu$.$