  A seesaw mechanism is presented in the neutrino sector and a new phase of CP
violation ($\alpha$) emerges in the interplay between the type-I and type-III
seesaw schemes. This phase is inside the mixing term, and thus it cannot be
rotated away in the Yukawa Lagrangian and, therefore, the heavy symmetry states
cannot be in a diagonal weak basis in the broken phase. Some particular
descriptions are analyzed suggesting that if the usual Yukawa couplings are
suppressed, leptogenesis still occurs due to a new interacting vertex with
fermion triplet $T$, fermion singlets $N$, and an ad-hoc scalar triplet,
$\Sigma$, which now is included to mediate the interactions. The evaluated CP
violation is enough to generate the observed matter-antimatter asymmetry even
in the minimal $1N+1T$ case (independently of $\alpha$) or in the $2N+1T$
approach (controlled by $\alpha$). The latter introduces more CP contributions
to leptogenesis due to new diagrams which are now possible even with the
suppressed imaginary part of the standard Yukawa couplings and can induce the
observed baryon-to-photon ratio.
