C. F. Keil and Franz Delitzsch on Psalms 108:1

Two Elohimic Fragments Brought Together

The אודך in Psa 108:4 and the whole contents of this Psalm is the echo to the הודוּ of the preceding Psalm. It is inscribed a Psalm-song by David, but only because it is compiled out of ancient Davidic materials. The fact of the absence of the למנצח makes it natural to suppose that it is of later origin. Two Davidic Psalm-pieces in the Elohimic style are here, with trifling variations, just put together, not soldered together, and taken out of their original historical connection.

That a poet like David would thus compile a third out of two of his own songs (Hengstenberg) is not conceivable.