EDIT: The original question was "Is a linear algebraic group with finite maximal compact subgroup finite?". This was answered by Tsemo Aristide. The answer is NO. $\mathbb{R}$ is a linear algebraic group and its maximal compact subgroup is finite, $\{-1,+1\}$.
Maybe you can still help me to show that my specific linear algebraic group is finite.
I have a linear algebraic group $G \subset GL(\mathbb{C}, d)^{\otimes n}$ for which I could show that it contains only finitely many unitaries. Moreover, I could already show that each component of $G$ contains at most one unitary. Components that do not contain a unitary, are composed of positive-definite matrices. Also the identity component contains only positive-definite matrices. Maybe this can be used to show that $G$ is finite.
I can also give you more information on the group $G$ if this is necessary to answer the questions above. However, this will require many details.
Thank you for your help!
No, $\mathbb{R}$ is a linear algebraic group and its maximal compact subgroup is finite.