  In this paper, we present and characterize the injective hull of a
two-Lipschitz operator ideals and the definition of two-Lipschitz dual operator
ideal. Also we introduce two methods for creating ideals of two-Lipschitz
operators from a pair of Lipschitz operator ideals. Namely, Lipschitzization
and factorization method. We show the closedness, the injectivity and the
symmetry of these two-Lipschitz ideals according to the closedness, injectivity
and symmetry of the corresponding Lipschitz operator ideals. Some illustrative
examples are given.
