  We show that the rank decomposition of a real matrix $r$, which is a
$Spin(3,1)$ tensor, leads to $2k$ Majorana bispinors, where $k= rank\: r$. The
Majorana bispinors are determined up to local $GL(k, \mathbb{R})$
transformations. The bispinors are combined in pairs to form $k$ complex Dirac
fields. We analyze in detail the case $k=1$, in which there is just one Dirac
field with the standard Lagrangian. The $GL(1, \mathbb{R})$ symmetry gives rise
to a new conserved current, different from the well known $U(1)$ current. The
$U(1)$ symmetry is present too. All global continuous internal symmetries in
the $k=1$ case form the $SO(2,1)$ group. As a side result, we clarify the
discussed in literature issue whether there exist algebraic constraints for the
matrix $r$ which would be equivalent to the condition $rank\: r=1$.
