Talk:SU(2)

Is there a reference to be had for why the special unitary group is of dimension n^2 - 1 over the reals? Though correct, the number of equations given by the unitary condition and the det = 1 (again over R) is n^{2}-2, and then turns out one of the equations is dependent after checking the Jacobian at the identity. But would like a uniform way of seeing this.