# Clebsch-Gordan coefficents necessary and sufficent condition to be non-zero

I know that the Clebsch-Gordan coefficient, $$\left<J_1, m_1, J_2, m_2|J,M\right>,$$ is zero if the following conditions are not satisfied: $$|J_1-J_2| \le J \le J_1+J_2,$$ $$m_1+m_2=M,$$ $$|M| \le J.$$ My question is whether it is possible that a Clebsch-Gordan coefficient satisfies these conditions but is still zero? I.e. are these conditions been satisfied a necessary and sufficient condition for a non-zero Clebsch-Gordan coefficient or only a necessary? and can it be proved either way?

A Clebsch-Gordan coefficient can be zero even if those conditions are satisfied. For example*, using the notations $|J, M\rangle$ and $|J_1, m_1, \, J_2, m_2 \rangle$: $$|\,2, 0\,\rangle= \frac{1}{\sqrt{2}}|\,2,+1,\,1,-1\,\rangle - \frac{1}{\sqrt{2}}|\,2,-1,\,1,+1\,\rangle \,,$$ so that there is no projection on the $m_1=m_2=0$ state: $$\langle J, M |J_1, m_1,\, J_2, m_2 \rangle = \langle \,2,0\,| \,2,0,\,1,0\, \rangle = 0\,.$$