Let $$\mathbb{R}$$ denote the set of all real numbers and let $$i=\sqrt{-1}$$. Consider the matrices

$$S=\begin{bmatrix}0&-1\\1&0\end{bmatrix}\quad\text{and}\quad T=\begin{bmatrix}1&1\\0&1\end{bmatrix}.$$

Let $$a,b,c,d$$ be real numbers such that

$$ST=\begin{bmatrix}a&b\\c&d\end{bmatrix}.$$

Let

$$H=\{\,x+iy:\;x,y\in\mathbb{R}\;\text{and}\;y>0\,\}.$$

Then which of the following statements is (are) TRUE?