There's two problems here. One is convincing people that you're smart and not just acting smart. The mistake is assuming that people can recognize directly that someone else is smarter than them. The more effective way to be recognized as smart is social proof.
The second problem is that being smart is a kind of social power (like being rich or beautiful) so there is an incentive not to recognize your intelligence if it's seen as a social threat. Showing someone that they're wrong usually results in a loss of social value for them. Something like sharing your IQ score can be seen as an unprovoked signal of social power.
If you want to leverage your intelligence politically then the most important thing is to signal your political allegiances first. Generally power is recognized when others realize that it's in their interest to do so.
The second problem is that being smart is a kind of social power (like being rich or beautiful) so there is an incentive not to recognize your intelligence if it's seen as a social threat. Showing someone that they're wrong usually results in a loss of social value for them. Something like sharing your IQ score can be seen as an unprovoked signal of social power.
If you want to leverage your intelligence politically then the most important thing is to signal your political allegiances first. Generally power is recognized when others realize that it's in their interest to do so.