Having stated the behaviour of the covariant derivatives in respect of the variations caused by an infinitesimal coordinate transformation, we obtain a simple expression of the energy-momentum tensor for a field defined, in a riemannian space-time, by a second-rank antisymmetric tensor. A field defined by a second-rank symmetric spinor and by his complex conjugate, in a riemannian space-time, is then considered and, by the invariance of the action for a general infinitesimal coordinate transformation and also for an infinitesimal local Lorentz transformation, the expression of the divergenceless energy-momentum spinor is deduced. Einstein—Maxwell equations in spinor form are finally derived from a variational principle, when a source-free electromagnetic field is supposed in presence of the gravitational field.