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.