And a more natural clustering would reflect that.
What subspace are you doing your clustering in, though? Both the pro-capital-punishment and anti-capital-punishment side should be able to agree that capital punishment and “central” murder are similar in the “intentional killing of a human” aspects, but differ in the “motives and decision mechanism of the killer” aspects (where the “central” murderer is an individual, rather than a judicial institution). Each side has an incentive to try to bind the murder codeword in their shared language to a subspace that makes their own side’s preferred policy look natural.
What subspace are you doing your clustering in, though? Both the pro-capital-punishment and anti-capital-punishment side should be able to agree that capital punishment and “central” murder are similar in the “intentional killing of a human” aspects, but differ in the “motives and decision mechanism of the killer” aspects (where the “central” murderer is an individual, rather than a judicial institution). Each side has an incentive to try to bind the murder codeword in their shared language to a subspace that makes their own side’s preferred policy look natural.