Q. Is it Chicago style to not capitalize clauses in the US Constitution (the commerce clause, the due process clause, etc.)? If so, what is the reasoning?
A. Such terms are not normally capitalized in Chicago style; see CMOS 8.81, which includes the example “the due process clause.” The reason for lowercase is that the Constitution doesn’t have a Commerce Clause or a Due Process Clause—not with those titles anyway.
Instead, “commerce clause” and “due process clause” refer to certain passages in the Constitution that treat commerce and due process. We realize, however, that those terms (and others naming specific clauses) have acquired the status of proper nouns for many writers and are often styled with initial caps in published prose. And Merriam-Webster, though it uses lowercase in its headwords for such terms (see, e.g., the entry for “due process clause”), adds the label “often capitalized.”
If Chicago’s default style is too conservative for you, capitalization-wise, there’s nothing wrong with applying initial caps to such terms as long as you stick to clauses (e.g., the Due Process Clause, but the legal concept of due process).