Let $(X, \omega)$ be a compact Kähler manifold. The Kähler identities express the commutator relations between the operators $$\partial, \ \ \overline{\partial}, \ \ L,$$ and their adjoints. To be clear, $L : \Lambda^k \to \Lambda^{k+2}$ is the Lefschetz operator $\alpha \mapsto \alpha \wedge \omega$.
Although the identities are not hard to prove, remembering them is a challenge that I am yet to conquer. This leads me to ask whether there is a more enlightened perspective to be had when it comes to the Kähler identities.
Edit: This is also discussed in the following MO post: https://mathoverflow.net/questions/64520/global-algebraic-proof-of-the-kahler-identities
For the sake of completeness, let me post an answer. As alluded to here, the Kähler identities are local. Therefore, an algebraic proof largely misses the point. In more detail, the Kähler identities fundamentally use the fact that a Kähler metric affords coordinates in which it coincides with the Euclidean metric up to second-order. Hence, a coordinate-free, or algebraic proof would, at best, mask this property.