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.

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