I'll echo Ben Webster's comment from the other thread, but in the other direction: find some natural commuting operators whose simultaneous eigenspaces are all one-dimensional. This is how, for example, one gets the root space decomposition in Lie theory and the decomposition of spaces of modular forms into eigenforms. (Admittedly, the former is only unique up to a choice of Cartan subalgebra.)
Qiaochu Yuan
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