$a = e_1\wedge e_2 + e_3\wedge e_4$ is not decomposable

The answer follows in the sentence after what you are looking at:

(This is a symplectic form, since α ∧ α ≠ 0.)

Meaning that if $a$ could be written as a single wedge product of $2$ vectors, then its wedge-square would be zero.

But $a\wedge a=(e_1 \wedge e_2 + e_3\wedge e_4)\wedge(e_1 \wedge e_2 + e_3\wedge e_4)=\\e_1\wedge e_2\wedge e_3\wedge e_4+e_3\wedge e_4\wedge e_1\wedge e_2=2(e_1\wedge e_2\wedge e_3\wedge e_4)\neq 0$