Yes it is (see Sakuraei).

Dear Behzadi

thank you for your reply, please make me clear what is it possible where we have to coupled angular momentum operators, [J,K] is non vanishing!!!!!?

In other word  based on the addition theorem (Sakuraei) we emphasis that [J,K] =0.

Let us assume that we are adding angular momentum operators J1 and J2. For example we can add orbital angular momentum with spin angular momentum. Then components of J1 and J2 satisfy  [J1i, J1j]= εijk J1k and [J2i, J2j]= εijk J2k. These commutators do not vanish, but as you have rightly noticed, all components of  J1 and J2 commute between each other. 

We define vector sum of operators J= J1 + J2. You can check by direct substitution that components of J also satisfy [Ji, Jj]= εijk Jk. In this way we can say that J is also a legitimate angular momentum operator.

Dear Biswajoy Brahmachari 

Thank you for your reply, I wounder if I ask you to determine what is the angular momentum addition theorem possible where we have to coupled angular momentum operators, [J,K] is non-vanishing!!!!!?

No. I can tell you only about the case when J1 and J2 commute. That is they are simultaneously measurable quantities.

Suppose we have two spin half particles. When they combine to form a bound sate, respective angular momenta of individual particles can form many different types of combinations. Each possible type of combination represents a quantum state. These states then can be grouped in singlet, triplets etc. In this case we can combine states symmetrically to form a triplet of combinations, or we can also combine anti symmetrically to form a  singlet of combinations. Therefore in this case (inside the bound state) four types of different spin combinations are possible. i) uu ii) dd iii) 1/√2 [ud+du] iv) 1/√2 [ud-du]. The first three can be written as |1,1>, |1,-1> and |1,0> in the |j,m> basis and the fourth can be written as |0,0>.

Numerical coefficients which appear during the combination process, when we normalize eigenstates, are known as Clebsch-Gordon coefficients. In our case for the symmetric combination, we see that CG coefficients are +1/√2 and +1/√2. Whereas for the antisymmetric case CG coefficients  are +1/√2 and -1/√2. Using these coefficients we can transform |m1,m2> basis into |j,m> basis. Maximum allowed value of j is j1+j2 whereas the minimum allowed value is |j1-j2|. In our case maximum is 1/2+1/2=1 whereas minimum is 1/2-1/2=0.

Do you need more detailed information on this topic? Ofcourse as I have said, I can tell you only about the case when J1 and J2 commute.

Ref. Quantum Mechanics by Powell and Crasemann, Sec 10.10 and Sec 10.12

Thank you for your replies.