  By making use of the decomposition theory of gauge potential, the inner
structure of SU(2) and SO(4) gauge theory is discussed in detail. We find the
SO(4) monopole can be given via projecting the SO(4) gauge field onto an
antisymmetric tensor. This projection fix the coset $% SU(2)/U(1)\bigotimes
SU(2)/U(1)$ of SO(4) gauge group. The generalized Hopf map is given via a Dirac
spinor. Further we prove that this monopole can be consider as a new
topological invariant. Which is composed of two monopole structures. Local
topological structure of the SO(4) monopole is discussed in detail, which is
quantized by winding number. The Hopf indices and Brouwer degree labels the
local property of the monopoles.
