X=X^*
\qquad
\text{for}
\quad
X\in (M_{\mu\nu}, P_\alpha)\end{gathered}$$ since the relations ,  are invariant with respect to such conjugation[^3]. It is well-known that real structures defined in this way are in one-to-one correspondence with the metric signatures $(p, q)$, however complexified algebra is signature independent.

It has been found in [@Zakrz-Comm] that for any (non-zero) vector $\tau=\tau^{\mu}e_{\mu}\in V$ (and any metric tensor $g_{\mu\nu}$ as above) one can introduce the corresponding classical $r$-matrix $$\begin{gathered}
r_{(\tau,g)}={\tau}^{\alpha}M_{\alpha \mu}\wedge P^{\mu}\equiv {\tau}^{\alpha}g^{\beta\sigma}M_{\alpha \beta}\wedge
P_{\sigma} \equiv {\frac{1}{2}}\tau\llcorner \Omega_g \in \wedge^{2}\operatorname{iso}(g),
$$ where $\Omega_g=M_{\mu \nu}\wedge P^{\mu}\wedge P^{\nu}$ is known to be the only invariant element in $\wedge^{3}\operatorname{iso}(g)$ and $\tau \llcorner$ is used for contraction with the vector $\tau$. The Schouten bracket reads $$\begin{gathered}
[[r_{(\tau,g)},r_{(\tau,g)}]]=- \tau_g^2\Omega_g,
\label{z2}\end{gathered}$$ where $\tau_g^2\equiv \tau^\mu\tau_\mu\equiv g_{\mu\nu}\tau^\mu\tau^\nu$ denotes the scalar square of $\tau$ with respect to the metric $g$. One should notice that the case $\tau_g^2=0$, which is only possible for non-Euclidean signature, provides a solution of the classical (non-modified) Yang–Baxter equation. Further on we shall simplify the notation and drop off the sub-index referring to the metric $g$.

We are now in position to introduce the corresponding quantized Hopf algebra structure. According to [@Zakrz-Comm] non-equivalent quantizations are classified by the conjugation classes of the 