We will construct the exceptional isomorphism .
The group acts on preserving the usual pairing where the are the usual basis vectors.
There is an invariant line , the span of and an invariant hyperplane . Let . acts on .
Since is the radical of the pairing on , the pairing descends to a non-degenerate bilinear pairing on . As it is symmetric and we are in characteristic 2, it is automatically skew-symmetric.
The -action preserves this pairing, hence we get our desired homomorphism from to .
To check injectivity, it suffices to show that is not in the kernel, since we know all normal subgroups of . Surjectivity then follows by a counting argument, so we get our desired isomorphism.