$V$ is an inner product space of finite dimension over $\mathbb{R}$, and $T:V\to V$ a linear transformation which is normal, that is, $T^\*T=TT^\*$. In addition $T^2=T$. Prove $T$ is self adjoint, that is, $T=T^\*$.

I tried to prove it algebraically by using the inner product but it didn't work for me. Then I tried to prove the statement that $T$ is self adjoint iff $\langle v,Tv\rangle$ is real for all $v$.

I know that $T$ is diagonalizable, therefore there is a basis of $V$ consisting of eigenvectors. In addition every eigenvector of $T$ is an eigenvector of $T^\*$. Can I prove that $\langle v,Tv\rangle$ is real only for the vectors in the basis and then it means it is for all $v$? 

Any help or further hints are very appreciated.