David signed this painting in two places; he put his full signature under Crito, the young man clutching Socrates's thigh, and his initials under Plato. David's signature placement often had symbolic meaning – for example, in his painting of Count Potocki, David signed in the collar of the dog that is barking at the sitter. In Death of Socrates, his signatures also have meaning. His initials under Plato are a reference to the fact that the story comes from Plato, a thanks for the inspiration. His fuller signature under Crito means that this is the character whom the artist identifies most with. This may be a reference for Crito’s position in the composition – clutching Socrates's thigh. In this way, David would be seen as a man who likewise clutches at the morals and values that Socrates represents.
There is some controversy as to the origins of the painting. Vidal states that "the younger of Trudaine de Montigny's (fr) two sons commissioned David's Death of Socrates in 1786",[4] but Bordes says that "the drawing for Socrates…bears the unexpected date of 1782." However, it is possible that the commission in 1786 was for a painting that David had already conceived as early as 1782, and Bordes does acknowledge that the date is an addition to the drawing.[5] Regardless of when it was originally conceived, the painting was completed in 1787, in Paris.