+ // cyclicity without gamma5
+ e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
+ - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(mu);
+ e = dirac_trace(e);
+ result += check_equal(e, 0);
+
+ e = dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam)
+ - dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(lam) * dirac_gamma(mu);
+ e = dirac_trace(e).expand();
+ result += check_equal(e, 0);
+
+ // cyclicity of gamma5 * S_4
+ e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig)
+ - dirac_gamma(sig) * dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho);
+ e = dirac_trace(e);
+ result += check_equal(e, 0);
+
+ // non-cyclicity of order D-4 of gamma5 * S_6
+ e = dirac_gamma5() * dirac_gamma(mu) * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap) * dirac_gamma(mu.toggle_variance())
+ + dim * dirac_gamma5() * dirac_gamma(nu) * dirac_gamma(rho) * dirac_gamma(sig) * dirac_gamma(kap);
+ e = dirac_trace(e);
+ e = (e / (dim - 4)).normal();
+ result += check_equal(e, 8 * eps0123(nu, rho, sig, kap));
+
+ // one-loop vacuum polarization in QED
+ e = dirac_gamma(mu) *
+ (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
+ dirac_gamma(mu.toggle_variance()) *
+ (dirac_slash(l, dim) + m * dirac_ONE());
+ e = dirac_trace(e).simplify_indexed(sp);
+ result += check_equal(e, 4*((2-dim)*l*l + (2-dim)*ldotq + dim*m*m).expand());
+
+ e = dirac_slash(q, dim) *
+ (dirac_slash(l, dim) + dirac_slash(q, dim) + m * dirac_ONE()) *
+ dirac_slash(q, dim) *
+ (dirac_slash(l, dim) + m * dirac_ONE());
+ e = dirac_trace(e).simplify_indexed(sp);
+ result += check_equal(e, 4*(2*ldotq*ldotq + q*q*ldotq - q*q*l*l + q*q*m*m).expand());
+