The components of Dirac Equation -- bosonic Lagrangian?

[arivero]

The four components of Dirac equations obey the Klein-Gordon equation for a particle of mass m. This is always explained when introducing Dirac equation, but it is never exploited further. I am wondering:

• Can we then write a bosonic lagrangian for these four "particles"?
• Is this related to the existence of supersymmetry? We can extract the components for a Dirac equation in any number of dimensions, but supersymmetry only exists for D=10, 6, 4, 3 if I recall correctly.
• Does the process generalise to charged fermions?