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We construct firstly the complete list of five quantum deformations of $D=4$complex homogeneous orthogonal Lie algebra $\mathfrak{o}(4;\mathbb{C})\cong\mathfrak{o}(3;\mathbb{C})\oplus \mathfrak{o}(3;\mathbb{C})$, describingquantum rotational symmetry of four-dimensional complex space-time, inparticular we provide the corresponding universal quantum $R$-matrices. Furtherapplying four possible reality conditions we obtain all sixteen Hopf-algebraicquantum 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 bythe 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.

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