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We employ new calculational technique and present complete list of classical$r$-matrices for $D=4$ complex homogeneous orthogonal Lie algebra$\mathfrak{o}(4;\mathbb{C})$, the rotational symmetry of four-dimensionalcomplex space-time. Further applying reality conditions we obtain the classical$r$-matrices for all possible real forms of $\mathfrak{o}(4;\mathbb{C})$:Euclidean $\mathfrak{o}(4)$, Lorentz $\mathfrak{o}(3,1)$, Kleinian$\mathfrak{o}(2,2)$ and quaternionic $\mathfrak{o}^{\star}(4)$ Lie algebras.For $\mathfrak{o}(3,1)$ we get known four classical $D=4$ Lorentz $r$-matrices,but for other real Lie algebras (Euclidean, Kleinian, quaternionic) we providenew results and mention some applications.

Quantum deformations of D=4 Euclidean, Lorentz, Kleinian and quaternionic o^*(4) symmetries in unified o(4;C) setting