  Calculation of band structure of three dimensional photonic crystals amounts
to solving large-scale Maxwell eigenvalue problems, which are notoriously
challenging due to high multiplicity of zero eigenvalue. In this paper, we try
to address this problem in such a broad context that band structure of three
dimensional isotropic photonic crystals with all 14 Bravais lattices can be
efficiently computed in a unified framework. We uncover the delicate machinery
behind several key results of our work and on the basis of this new
understanding we drastically simplify the derivations, proofs and arguments in
our framework. In this work particular effort is made on reformulating the
Bloch boundary condition for all 14 Bravais lattices in the redefined
orthogonal coordinate system, and establishing eigen-decomposition of discrete
partial derivative operators by systematic use of commutativity among them,
which has been overlooked previously, and reducing eigen-decomposition of
double-curl operator to the canonical form of a 3x3 complex skew-symmetric
matrix under unitary congruence. With the validity of the novel nullspace free
method in the broad context, we perform some calculations on one benchmark
system to demonstrate the accuracy and efficiency of our algorithm.
