We study the alpha-alpha bremsstrahlung using a generalization of the Siegert theorem. The corresponding transition amplitude is expressed through the electric and magnetic field strengths and matrix elements of the generalized electric and magnetic dipole moments of the system. Therefore, the amplitude has a manifestly gauge-independent form. Special attention is paid to taking into account the interplay between the Coulomb repulsion and strong interaction of colliding alpha particles. The bremsstrahlung cross sections obtained are compared to available experimental data.