On page 350 of Hall's book real roots of $U(2)$ are listed as $(1, 1)$ and $(-1, -1)$ after identifying the maximal torus algebra $\frak{t}$ of diagonal matrices with $\mathbb{R}^2$. However, my calculations show that the roots should be $(1, -1)$ and $(-1, 1)$, and I don't see what am I missing. Here are the calculations:
Let $H=\text{diag}(ai, bi)$ and $$X_1=\begin{bmatrix}0&1\\-1&0\end{bmatrix}, X_2=\begin{bmatrix}0&i\\i&0\end{bmatrix}.$$ Then $[H, X_1]=(a-b)X_2$ and $[H, X_2]=(b-a)X_1$ which implies that $$[H, X_1+iX_2]=-i(a-b)(X_1+iX_2), \quad [H, X_1-iX_2]=i(a-b)(X_1-iX_2).$$ Thus, the real root corresponding to $X+iX_2\in \mathfrak{t}_\mathbb{C}$ is $\alpha=\text{diag}(-i, i)\in \mathfrak{t}$ because $$\langle \alpha , H\rangle =\text{tr}(\alpha^*H)=b-a$$.
I suspect that we are using different inner products (because at the bottom of page 351 apparently the author gets $\langle \alpha, \alpha\rangle=1$.) Any clarification is appreciated.
I ran into the same problem earlier and my calculation was the same as yours. My understanding is that the author meant to say, the roots are $(1,-1)$ and $(-1,1)$. We select $\alpha = (1,-1)$ as our positive root. And decompose elements in $\mathfrak{t}$ as a $\mathbb{R}$-linear combination of $\alpha = (1,-1), \ \beta = (1,1)$. This would be coherent with the later conclusions.