Tanks a lot! Yes and Sorry, I forgot to call simplified_indexed. Still, I get some confused with the index variance. and excuse me to let me explain my options in some details. I understand g_c^d = g_{ce} g^{ed}, and let’s come back to the ret expresion: "auto ret = indexed(A, sy_anti(0,1,2,3), ia,ib,ic,id) *lorentz_g(ic, id.toggle_variance());” In my understanding, it corresponds to A^{abcd} * g^{c}_{d}, is that right? if so, then A^{abcd} * g^{c}_{d} = A^{abcc}, so according to sy_anti, it is 0, directly from GiNaC. So if I want to contract two index in A, I mean, we need this one: A^{abc}_{c} = A^{abcd} * g^{dc} ? Thanks very much! Best Regards! Feng