How did the snake die?

Cadmus embedded a spear in the snake's back and a sword in its throat, forcing the snake backward until the sword pinned the snake to a tree.

But the wound is slight, because the serpent draws back from the thrust, pulling its wounded neck away, and, conceding its wound, keeps back the sword, and does not let it sink deeper. But the son of Agenor following it all the time presses the embedded iron into its throat, until an oak-tree blocks its backward course and neck and tree are pinned together. The tree bends under the serpent’s weight and the trunk of the oak groans with the lashing of its tail.