The Kähler identities (sometimes known as the Hodge identities) are an important collection of relationships between operators on the exterior algebra of a Kähler manifold. These relationships generalise to hermitian manifolds, sections of hermitian holomorphic vector bundles, and many other situations.

I know that the notion of a Kähler metric was introduced by Kähler himself in 1933 and that the Kähler identities were first generalised to hermitian manifolds by Demailly in 1985 (although he mentions that the ideas were present in a paper by Griffiths in 1966).

Can anyone fill in the historical gap and tell me where or when the Kähler identities first appeared?