He's much better at teaching languages than he is at learning them, and his vocabulary in his native language is extensive. He tries to focus on relevant subjects: status, politics, law and custom. These lessons therefore end up interspersed with explanations of those things, delivered through his usual inventive use of limited Kaled'a'in.
Courts and judges both turn out to be concepts he is familiar with, although the Haighlei versions are a little different from what Lionstar may previously have heard of. Filtered through the barrier of their mutual vocabulary, the system looks something like this:
There is a role called lawkeeper. People of certain social classes have the ability to act as lawkeepers for ceremonial and traditional reasons—Sakshemar himself is one such. But most lawkeeper work is done by people of an appropriate caste whose actual job is to investigate crimes, find out who did them, and bring those people before a court.
In court, the lawkeeper presents the judge with the evidence and reasoning by which they concluded that the accused party did the crime. There are a lot of formalized phrasings for this exchange, many of which Sakshemar can rattle off by heart because his father has done a lot of work as a judge; it's a respectable occupation for a member of the nobility.
Any of the three parties—the judge, the lawkeeper, or the accused—can call for a Truthsayer at any point in the proceedings. Truthsayers are scarce enough that this can be ruinously expensive, so it's almost never done; but the possibility does a lot to keep the participants honest, because deliberately lying to a judge in court is a crime punishable by death, and deliberately lying as a judge in court carries a penalty which Sakshemar leaves unspecified due to vocabulary but implies is generally considered to be worse. Courts keep meticulous records of who said what at what time, so if someone does call for a Truthsayer, there's no weaseling out by conveniently forgetting your lies.
Normally, though, threats to call for a Truthsayer are merely alluded to, very rarely acted on. Instead, the judge hears out the lawkeeper's case, asks the accused for their side of the story, and comes to one of three conclusions: that the accused is innocent, that they are guilty, or that they are—there's a phrase for this but the words are unfamiliar and Sakshemar has trouble defining them: the gist seems to be that they did it, but under some mitigating circumstance such that instead of the usual punishment for their crime they should be treated more leniently. The example he gives is of a case his father saw, where a servant was stealing jewelry from their employer, but turned out to be spending the proceeds on expensive treatments for their child's rare illness; his father ruled that the employer should keep back half the servant's wages until the value of the stolen items was thereby repaid, but promised to personally cover the child's medical expenses.
Aside from these fascinating lessons on language, culture, law, and politics, Sakshemar also helps Lionstar test various possible sources of ink and paper. After a couple of days, they have some sheets of bark which can be semi-permanently marked by an ink made from charcoal dust mixed with water. Sakshemar is mostly pretty cheerful about the whole process; he likes working with his hands, and likes the accomplishment of having a need and creating tools and materials to fulfill that need out of resources found in their environment. He gets frustrated occasionally when one of their attempts doesn't work out, but when they finally have something that works he beams delightedly about it.