The sfar operations and reality conditions for the complex quantum algebra U q ( d ( 4 C ) ) providing real quantum algebras U,,(0(6k . k ) ) k = 0, I. 2.3 and Uq(su(3. 1)) are classified. S W x d and non-standud sex operations me considered. It appears that only four choices of real forms (one with 141 = 1, h e with q real) provide real Hopf algebra Uq(su(2, 2)) = Uq(0(4, 2)) describing D = 4 conformal quanNm alpbras. We show that only the antipod-extended Cartan-Weyi basis of Uq(sl(4; C)) permits to define real q-defomd D = 4 wnformal algebra generators. In order to obtain the real D = 4 Weyi algebra as Hopf subalgebra of Uq(sw(2. 2)) only the non-standard real forms can be employed.

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