(CNN) For the first time in over a decade, the Supreme Court is being asked to review the legality of the military commissions at the US Naval Station in Guantanamo Bay, Cuba.

The pair of cases that reached the court this week could determine the future of military tribunals set up in the wake of the September 11, 2001 terrorist attacks.

Each case has been tied up for several years in the lower courts.

The first case, al Bahlul v. United States, raises the question whether the commissions have the constitutional authority to try domestic offenses, or whether their jurisdiction is limited to violations of the international laws of war.

Ali Hamza Ahmad Suliman al Bahlul, has been held in Guantanamo Bay since January 2002.

A Yemini and an al Qaeda propagandist, al Bahlul was convicted on three charges that the government concedes are not international war crimes. On appeal, the DC federal appeals court threw out two of those charges, but it fractured on the third, affirming al Bahlul's conspiracy conviction without agreement among the judges as to the grounds.

In court papers, his lawyers say the issue is of "exceptional importance," the idea that is it constitutional to try wholly domestic federal crimes, such as conspiracy, in these special non-judicial trial chambers.

Steve Vladeck, professor of law at the University of Texas School of Law and CNN Supreme Court analyst, who has filed friend-of-the-court briefs in the lower courts in both cases, says that although the case seems technical, it is central to the constitutional question confronting the post-September 11 military commissions.

"The Supreme Court has endorsed military commission trials for international war crimes, but it's never decided whether commissions may also try domestic offenses — and has at least implicitly suggested that the Constitution might require such cases to be brought in civilian courts. This is a big deal because it's very difficult to tie individual detainees to specific acts of terrorism — and, as such, to international war crimes. That's why almost all of the charges (and convictions) before the post-September 11 commissions have involved more ordinary, domestic offenses, which the government can often prove without showing a direct connection to specific combat activities," he said.

The petition also challenges his conspiracy conviction on First Amendment grounds, and objects to the entire military commission system on the grounds that, because it is only targeted at non-citizens, it is unconstitutionally discriminatory under the Fifth Amendment's due process clause.

The second case, al Nashiri v. Obama, involves Abd al-Rahim al-Nashiri, one of the defendants currently before the commissions, who faces capital charges for his alleged role in the October 2000 bombing of the USS Cole and the 2002 bombing of a French tanker.

Nashiri is seeking to challenge the power of the commissions to try him for offenses that pre-dated the September 11 attack, since the statute authorizing military commissions limits their authority to crimes committed during "any conflict subject to the laws of war." Unless the United States was already at war with al Qaeda in October 2000, Nashiri argues, he can't be tried in a military (as opposed to civilian) court for his alleged role in the Cole bombing.

In Nashiri's case, the lower courts did not reach the merits of his challenge, holding that it was premature because Nashiri had not yet been convicted — and that such arguments are better presented on a post-conviction appeal. Nashiri argues, in response, that he is challenging the military commission's power to try him in the first place, a right that cannot fully be vindicated after the fact; and that, in any event, given how slowly the commissions have operated, and how harshly he has been treated while in US custody, it would be both unfair and inconsistent with prior Supreme Court decisions to hold off on resolving his challenge until after he is convicted.

Nashiri was seized by local authorities in Dubai in late 2002 and transferred to the custody of the Central Intelligence Agency. His brief filed with the court is blacked out on many pages and marked "unclassified for public release."

"One of the central issues in Nashiri's case is whether or not the torture he suffered weighs in favor of review at this time as opposed to after trial," said his lawyer, Rick Kammen.

As such the briefs details his time spent in the CIA's Rendition Detention and Interrogation Programs.

The Trump administration is likely to respond within 30 days. The justices are not expected to act on the petitions until this summer.