Recognizability of finite simple groups $L_4(2^m)$ and $U_4(2^m)$ by spectrum

It is proved that finite simple groups $L_4(2^m)$, $m\ge2$, and $U_4(2^m)$, $m\ge2$, are, up to isomorphism, recognized by spectra, i.e., sets of their element orders, in the class of finite groups. As a consequence the question on recognizability by spectrum is settled for all finite simple groups without elements of order 8.