Clebsch-Gordan ambiguity not sign-related?

Question

I'm learning to compute Clebsch-Gordan coefficients. For $j_1=1, j_2=1/2, j = 3/2$ and $m=1/2$ I got $$ |3/2, 1/2, 1, 1/2 \rangle = \sqrt{2/3} |1,0,1/2,1/2\rangle + \sqrt{1/3} |1,1,1/2,-1/2\rangle$$

using $J_-$ over $|3/2, 3/2, 1, 1/2\rangle$, where I followed the formalism $|j, m, j_1, j_2\rangle$ $\rightarrow$ $|j_1, m_1, j_2, m_2\rangle$. This result is correct following the PDG conventions.

For $m=1/2$ fixed, there is only another state possible and it is $|1/2, 1/2, 1, 1/2\rangle$. Using $J_+$ over $|1/2, -1/2, 1, 1/2\rangle$ I get $$|1/2, 1/2, 1, 1/2\rangle = \sqrt{2} |1,1,1/2,-1/2\rangle + |1,0, 1/2,1/2\rangle$$ which is not possible nor is it orthogonal. What is happening here?

Answer 1

Okay, I already saw my error. I'll post it here in case it helps someone.

That decomposition is in fact orthogonal if I normalize the state and then change one of the signs due to the ambiguity in the signs of the coefficients.