Symmetry of Second Derivatives - in Lie Theory

In Lie Theory

Consider the first-order differential operators Di to be infinitesimal operators on Euclidean space. That is, Di in a sense generates the one-parameter group of translations parallel to the xi-axis. These groups commute with each other, and therefore the infinitesimal generators do also; the Lie bracket

= 0

is this property's reflection. In other words, the Lie derivative of one coordinate with respect to another is zero.

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