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In the coffee hour discussion today, Nick gave me an interesting explicit example of an exceptional isomorphism of Lie groups.

is an isomorphism via the map. Let me elaborate. The group acts naturally on . If is a basis of , then a basis of could be taken as . Thus, is and our -action gives a monomorphism of groups:

.

The image must actually be inside .

The quadratic form

,

is symmetric (two permutations) and preserves the – action, i.e.,

.

Hence the image of in must also preserve this symmetric bilinear form. Thus, . By dimension consideration, they must be equal.

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