Thanks very for the reply.

and I have tried the suggestion, i.e., using the following expression:

auto ret = indexed(A, sy_anti(0,1,2,3), ia,ib,ic,id) * lorentz_g(ic, id.toggle_variance());

but I got something like: -delta.d~c*A~d~a~b~c, still a non-zero result.

BTW, if c’s variance is not toggled in lorentz_g, 

      then these two dummy index c’s variance is not matched, and is that allowed?

Thanks again!