the case of $\tau^2=0$, which includes the null-plane quantum Poincaré Hopf algebra [@Ballesteros_null1; @Ballesteros_null2; @Ballesteros_null3] (also known as the light-like deformation). Throughout this section we assume non-Euclidean signature.

We remind the $D=2+(D-2)$ orthogonal decomposition for inhomogeneous orthogonal Hopf algebra of non-Euclidean signature which relies on a suitable choice of basis in the vector space $V$. This in turn invokes change of the generators of $U(\operatorname{iso}(g))$ which are now related with the so-called “null-plane” basis in $\operatorname{iso}(g)$ $$\begin{gathered}
P_{\mu}= (P_{+},P_{-},P_{a} ),
\qquad
M_{\mu \nu}= (M_{+ -},M_{+ a},M_{- a},M_{ab} ),
\qquad
a,b=1,2,\ldots, D-2.
\label{lc1}\end{gathered}$$ Here we took two (non-orthogonal) null vectors $\tau^{\mu}\equiv \tau_{+}^{\mu}=(1,0,{\ldots}, 0)$, $\tilde{\tau}^{\mu}\equiv \tau_{-}^{\mu}=(0,1,0,{\ldots}, 0)$: $\tau_{+}\tau_{-}=1$ (as spanning of 2-dimensional Lorentzian subspace)[^11] in order to obtain the convenient null-plane (a.k.a.light-cone) basis in the space of the Lie algebra generators . This algebra consists of the following (non-vanishing) commutators $$\begin{aligned}
{3}
& [M_{+a},M_{-b}]=-i(M_{ab}+g_{ab}M_{+-}),
\qquad &&
[M_{\pm a},M_{\pm b}]=0,&
\label{M+aM-b}
\\
& [M_{\pm a},M_{bc}]=i(g_{ab}M_{\pm c}-g_{a c}M_{\pm b}),
\qquad &&
[M_{+-},M_{\pm a}]=\pm iM_{\pm a},
\label{MpmMbc}
\\
& [M_{+-},P_{\pm}]=\pm i P_{\pm},
\qquad &&
[M_{\pm a},P_{b}]=ig_{ab}P_{\pm}, &
\label{M_pmP}
\\
& [M_{\pm a},P_{\pm}]=[M_{+-},P_{a}]=0,
\qquad &&
[M_{\pm a},P_{\mp}]=-iP_{a} &
\label{M_pmP_pm}\end{aligned}$$ together with the standard commutation relations within the $(D-2)$-dimensional sector $(M_{ab}$, $P_{a},g_{ab})$, cf. , .

The universal 