Basic quantizations of $D=4$ Euclidean, Lorentz, Kleinian and quaternionic ${\mathfrak{o}}^{\star }(4)$ symmetries

We construct firstly the complete list of five quantum deformations of $D=4$ complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;$\mathbb{C}$)\cong \mathfrak{o}(3;$\mathbb{C}$)\oplus \mathfrak{o}(3;$\mathbb{C}$)$, describing quantum rotational symmetry of four-dimensional complex space-time, in particular we provide the corresponding universal quantum $R$-matrices. Further applying four possible reality conditions we obtain all sixteen Hopf-algebraic quantum deformations for the real forms of $\mathfrak{o}(4;\mathbb{C})$: Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian $\mathfrak{o}(2,2)$ and quaternionic ${\mathfrak{o}}^{\star }(4)$. For $\mathfrak{o}(3,1)$ we only recall well-known results obtained previously by the authors, but for other real Lie algebras (Euclidean, Kleinian, quaternionic) as well as for the complex Lie algebra $\mathfrak{o}(4;\mathbb{C})$ we present new results.