I am trying to show that, for a simple 2x2 complex matrix \begin{equation} U = \begin{bmatrix} a & b \\ c & d \end{bmatrix} \end{equation} there will be only 4 real constraints for $U$ to be unitary. I will denote the conjugate transpose by $\dagger$, for example the complex transpose of a matrix $A$ is $A^{\dagger}$. I will also use $*$ to denote complex conjugation of a complex number, for example $a^{*}$ is the complex conjugate of complex number $a$. I first carry out the calculation \begin{equation} UU^{\dagger}=\begin{bmatrix} a & b \\ c & d \end{bmatrix} \begin{bmatrix} a^{*} & c^{*} \\ b^{*} & d^{*} \end{bmatrix} = \begin{bmatrix} |a|^{2}+|b|^{2} & a c^{*} + b d^{*} \\ a^{*}c+b^{*}d & |c|^{2} + |d|^{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation} which leads to the constraints \begin{align} & |a|^{2}+|b|^{2} = 1 \\ & a c^{*} + b d^{*} = 0 \\ & a^{*}c+b^{*}d = 0 \\ & |c|^{2} + |d|^{2} = 1. \end{align} The second and third are same, and each of them gives 2 real constraints. The first and the fourth both give 1 real constraint. So in total the above four equations give 4 real constraints.
I know that $UU^{\dagger} = U^{\dagger}U$ and so $U^{\dagger}U=1$ should give no more constraints. But still I calculate it as \begin{equation} U^{\dagger}U= \begin{bmatrix} a^{*} & c^{*} \\ b^{*} & d^{*} \end{bmatrix}\begin{bmatrix} a & b \\ c & d \end{bmatrix} = \begin{bmatrix} |a|^{2}+|c|^{2} & a^{*} b + c^{*} d \\ ab^{*}+c d^{*} & |b|^{2} + |d|^{2} \end{bmatrix} = \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \end{equation} which leads to the constraints \begin{align} & |a|^{2}+|c|^{2} = 1 \\ & a^{*} b + c^{*} d = 0 \\ & ab^{*}+c d^{*} = 0 \\ &|b|^{2} + |d|^{2} = 1. \end{align} Now it seems to me that it is not obvious why the constraints obtained from $U^{\dagger}U = 1$ are independent of those obtained from $U U^{\dagger}=1$. However, we know that $U U^{\dagger}=1$ should imply $U^{\dagger}U = 1$. Could anyone give me some suggestions? I think I just miss some simple points, but currently haven't figured out the key points. Thanks a lot!
Since $ac^\ast=-bd^\ast$, $|a|^2|c|^2=|b|^2|d|^2$, i.e. $|a|^2-|a|^2|d|^2=|d|^2-|a|^2|d|^2$, whence $|a|=|d|$. But $|b|^2-|c|^2=|d|^2-|a|^2=0$, so$$ab^\ast=-bd^\ast\cdot(b/c)^\ast=-|b|^2(d/c)^\ast=-|c|^2(d/c)^\ast=-cd^\ast.$$