The last time the justices put on their robes and sat behind the Supreme Court’s majestic mahogany bench, Chief Justice John G. Roberts Jr. announced that former President Donald J. Trump enjoyed substantial constitutional immunity from prosecution.

That all happened on the first Monday in July. Fourteen weeks later, on the first Monday in October, the justices returned to the bench for the first argument of the new term. It concerned unemployment benefits.

It was a routine case, as most Supreme Court cases are, but it opened a window onto the justices’ mood and methods.

The question before them was intricate and technical: May people who sought unemployment benefits in Alabama during the coronavirus pandemic sue right away under a federal civil rights law over extreme delays or must they wait until the state’s Labor Department has rendered a final decision?

The Alabama Supreme Court ruled that the claimants could not sue until they had, in the legal jargon, exhausted their administrative remedies. The claimants called that decision “positively Kafkaesque.”

Except for a brief whispered aside between Justices Elena Kagan and Brett M. Kavanaugh that left them smiling, the justices were dour and diligent, methodically interrogating the lawyers before them. They were effectively starting their deliberations, discussing among themselves, for the first time, how the case should be resolved.