So, here's my guess based on the two chapters we have so far.
Lapis's younger sister was killed … by another commoner, and Lapis actually feels guilty for failing to protect her. The whole commoners being uncivilized attitude is something she adopted to deal with her pain, and so that she doesn't have to examine the actual situation too closely. After all, if commoners are ignorant and quick to anger, then something like this was probably unavoidable, nothing she could do.
As to why a commoner would kill her sister. In my experience (having grown up working class in a country where the class divide is still very much alive no matter what people say), enforcing the class divide can come from the bottom as well as the top. Working class people can get very annoyed by other working class people that have ideas and aspirations "above their station", and will starting talking about people getting "too big for their britches". Or I'm thinking to deeply and the killer was motivated by simple jealousy.