have many (non-isomorphic) incarnations, with the deformation parameter being both a formal variable (algebraic generator) or a numerical factor. The passage from the topological $h$-adic (Drinfel’d) version, we have used till now, to the $q$-analog (Drinfel’d–Jimbo) form and final specialization of the formal deformation parameter to some numerical value is well understood and described for standard deformations of semi-simple Lie algebras (see, e.g., [@ChP Chapter 9] and [@Klimyk Chapters 3, 7]). More deeper mathematical study of this problem can be found in [@Bonneau2; @Bonneau1]. The specialization problem for time-like $\kappa$-Poincaré case has been also treated in a bicrossproduct basis in the context of possible $C^*$-algebra reformulation [@Andrea; @Sitarz3; @Sitarz2; @Stachowiak].

In the present section we recall our earlier result concerning specialization problem for a time-like version of $\kappa$-type deformation [@BP2]. Then we extend the analysis to the non-standard (light-cone) case as well as to twisted $\kappa$-deformations. Surprisingly, to our best knowledge, the specialization problem for non-standard (i.e. invoked by a two-cocycle twist) deformation has not been studied in the literature yet. Our results show that (extended) Jordanian twist, in contrast to the Abelian one, enables (after suitable change of variables) to solve specialization problem.

The $\boldsymbol{D=1+(D-1)}$ orthogonal decomposition: $\boldsymbol{\tau^2\neq0}$
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