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It's well known that the automorphism group of the configuration of 27 lines on a smooth cubic surface in $\mathbb{P}^3$ (over a field containing all 27 lines) is isomorphic to the Coxeter group of type $ E_6 $.

Labelling the vertices of $ E_6 $ as $ x_1, x_2, x_3, x_4, x_5, y$ (so that $y$ is the unique leaf neighbouring the degree 3 node $x_3$) and labelling the 27 lines coming from looking at blowing up $ \mathbb{P}^2 $ at 6 points $ P_i $ in general position (and then making the anticanonical embedding) as $ e_i $ (exceptional lines of the blowup), $ l_{i,j} $ (proper transforms of lines through pairs of blown-up points) and $ c_j $ (proper transforms of conics going through all but one point), it's claimed in Hartshorne (V, Exercise 4.11) that one possible isomorphism between the automorphism group of the 27 lines and the Coxeter group of type $ E_6 $ is given by:
$x_1 \mapsto (e_1 \leftrightarrow e_2)$
$ \qquad \quad \vdots $
$x_5 \mapsto (e_5 \leftrightarrow e_6)$
and that $y$ is associated with the quadratic transformation of $ \mathbb{P}^2 $ based at the points $P_1, P_2, P_3$. (This is the birational self-map of $ \mathbb{P}^2 $ obtained by blowing up $P_1,P_2,P_3$, and then blowing down the proper transforms of the lines between them).

My question: how exactly does this birational transformation of $ \mathbb{P}^2$ induce an automorphism of the 27 lines, and how does one go about writing it down explicitly in terms of the $e_i$, $l_{i,j}$, $c_j$?

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