Determining the compact roots of the Cartan subalgebras of $\mathfrak sp(2,\mathbb R)$

Question

I want to understand the notions of real vs imaginary roots and compact vs noncompact roots (among the imaginary ones) in the theory of Cartan subalgebras (CSA's) of real semisimple Lie algebras. I consider the following example. Let $\mathfrak g=\mathfrak sp(2,\mathbb R) = \{X \in M(4,\mathbb R)| JX + ^t X J = 0\}$ the symplectic Lie algebra of rank $2$ with Cartan decomposition $\mathfrak g = \mathfrak k \oplus \mathfrak p$, where $\mathfrak k$ are the skew-symmetric matrices in $\mathfrak g$ and $\mathfrak p$ the symmetric ones in $\mathfrak g$. Now there are four classes of CSA's. Among them consider the maximally compact $\mathfrak b = \{\begin{pmatrix}0&0&\theta_1 &0\\0&0&0&\theta_2\\-\theta_1&0&0&0\\0&-\theta_2&0&0 \end{pmatrix} | \theta_1,\theta_2 \in \mathbb R\}$. Since this is compact, all roots of $(\mathfrak b^\mathbb C, \mathfrak g^\mathbb C)$ are imaginary. I want to see which are compact resp. noncompact. My questions: How does $\mathfrak b^\mathbb C$ look like? Isnt this just the set of matrices that look like $\mathfrak b$ but with complex entries? In Knapp, Representation theory of semisimple groups, page 404, he defines functionals $e_1,e_2$ on $\mathfrak b^\mathbb C$ giving $i \theta_1$ resp. $i \theta_2$. I do not understand how this is meant. (It cannot be meant that $e_1,e_2$ are extended by $\mathbb C$-linearity, since then we would not get purely imaginary values of the roots on $\mathfrak b^\mathbb C$.) Further I then want to understand why $\pm (e_1-e_2)$ are compact and the other ones noncompact. What would be a root vector for e.g. $\pm (e_1-e_2)$? All roots are given by $\pm e_1 \pm e_2, \pm 2 e_1, \pm 2 e_2$.