First, let us consider the blow-up of $\mathbb{P}^1_{\mathbb{C}} \times \mathbb{P}^1_{\mathbb{C}}$ at three points $\{(0, 0), (1, 1), (\infty, \infty)\}$.
It is a del Pezzo surface of degree $5$, and there are $10$ lines ( $(-1)$-curves), i. e., the three exceptional divisors $E_0, E_1, E_{\infty}$ corresponding to $(0, 0), (1, 1), (\infty, \infty)$, and the strict transforms of the seven curves $x = 0, x= 1, x= \infty, y = 0, y = 1, y = \infty, x = y$.
Next, I want to consider the blow-up of $\mathbb{P}^1_{\mathbb{C}} \times \mathbb{P}^1_{\mathbb{C}}$ at four points $\{(0, 0), (1, 1), (\infty, \infty), (p_1, p_2)\}$, where $p_1 \neq p_2$.
It sould be a del Pezzo surface of degree $4$, and I found the 13 lines, which are the four exceptional divisors corresponding to these four points, the strict transformations of the rulings $x = 0, x= 1, x= \infty, x= p_1, y = 0, y = 1, y = \infty, y = p_2$, and that of the diagonal line $x = y$.
However, I haven't found remained $3$ lines. So, my question is
Where are these remained $3$ lines? Can we get the defining equations of them?