${\rm ISO}(g)$ consists of ${\frac{1}{2}}D(D+1)$ generators $(M_{\mu\nu}, P_\alpha)$ adapted to a choice of the basis and satisfying the standard commutation relations $$\begin{gathered}
[M_{\mu \nu},M_{\rho \lambda}]=i(g_{\mu \lambda}M_{\nu \rho}-g_{\nu \lambda}M_{\mu \rho}+g_{\nu \rho}M_{\mu
\lambda}-g_{\mu \rho}M_{\nu \lambda}),
\label{MM}
\\
[M_{\mu \nu},P_{\rho}]=i(g_{\nu \rho}P_{\mu}-g_{\mu \rho}P_{\nu}),
\qquad
[P_{\mu},P_{\lambda}]=0.
\label{PP}\end{gathered}$$ The relation with the basis $\{e_{\mu}\}_{\mu=0}^{D-1}$ of $V$ is throughout the complexify vector representation[^2] $$\begin{gathered}
\label{rep}
M_{\mu\nu}\mapsto -i (g_{\mu\alpha}e_\nu-g_{\nu\alpha}e_\mu)\otimes e^\alpha \in \operatorname{End}V\otimes \mathbb{C}\end{gathered}$$ acting in the complexified vector space $V\otimes \mathbb{C}$. In fact, the generators $(M_{\mu\nu}, P_\alpha)$ belong to the complexified Lie algebra $\operatorname{iso}(g)$. However, for the purpose of this paper we shall treat the relations , as generating ones for a complex universal enveloping algebra $U(\operatorname{iso}(g))$ of $\operatorname{iso}(g)$ understood as a free (complex) unital associative algebra generated by the symbols $(M_{\mu\nu}, P_\alpha)$ and factorized further by a two-sided ideal generated by the relations , . It is also customary to consider real algebras as complex ones equipped additionally with the structure of involutive anti-linear anti-automorphism, the so-called $*$-conjugation: $X\mapsto X^*$, i.e. having the same formal properties as a Hermitean conjugation. In our case the choice of real structure is completely determined by the requirement that the generators ,  are self-conjugated (formally Hermitean or self-adjoint), i.e. $$\begin{gathered}
\label{RC1}
