  Using the isomorphism
$\mathfrak{o}(3;\mathbb{C})\simeq\mathfrak{sl}(2;\mathbb{C})$ we develop a new
simple algebraic technique for complete classification of quantum deformations
(the classical $r$-matrices) for real forms $\mathfrak{o}(3)$ and
$\mathfrak{o}(2,1)$ of the complex Lie algebra $\mathfrak{o}(3;\mathbb{C})$ in
terms of real forms of $\mathfrak{sl}(2;\mathbb{C})$: $\mathfrak{su}(2)$,
$\mathfrak{su}(1,1)$ and $\mathfrak{sl}(2;\mathbb{R})$. We prove that the $D=3$
Lorentz symmetry
$\mathfrak{o}(2,1)\simeq\mathfrak{su}(1,1)\simeq\mathfrak{sl}(2;\mathbb{R})$
has three different Hopf-algebraic quantum deformations which are expressed in
the simplest way by two standard $\mathfrak{su}(1,1)$ and
$\mathfrak{sl}(2;\mathbb{R})$ $q$-analogs and by simple Jordanian
$\mathfrak{sl}(2;\mathbb{R})$ twist deformations. These quantizations are
presented in terms of the quantum Cartan-Weyl generators for the quantized
algebras $\mathfrak{su}(1,1)$ and $\mathfrak{sl}(2;\mathbb{R})$ as well as in
terms of quantum Cartesian generators for the quantized algebra
$\mathfrak{o}(2,1)$. Finaly, some applications of the deformed $D=3$ Lorentz
symmetry are mentioned.
