An example from algebra is Albert's paper "On the Wedderburn norm condition for cyclic algebras" relating a 6-dimensional quadratic form to every biquaternion algebra (which is now known as the "Albert quadratic form"). His original paper is essentially one long (but of course extremely clever) computation. By now, we have much more conceptual proofs and we understand the situation much better (see for instance the Book of Involutions). I'm sure that there are many more examples of this style.
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