  The mathematical structure of the Born-Infeld field equations was analyzed
from the point of view of the symmetries. To this end, the field equations were
written in the most compact form by means of quaternionic operators constructed
according to all the symmetries of the theory, including the extension to a
non-commutative structure. The quaternionic structure of the phase space was
explicitly derived and described from the Hamiltonian point of view, and the
analogy between the BI theory and the Maxwell (linear) electrodynamics in
curved space-time was explicitly shown. Our results agree with the observation
of Gibbons and Rasheed that there exists a discrete symmetry in the structure
of the field equations that is unique in the case of the Born-Infeld nonlinear
electrodynamics.
