How to prove the adjoint of this operator is the $-T-I$

Question

Suppose $\mathbb R$ is a finite-dimensional real inner product space, and $T$ is an operator on it with the following properties

$1. T^2=T^*$

$2. T$ is invertible

$3. T-I$ is invertible

Prove that $T^*=-T-I$

I don't know how to approach this problem. In particular, I don't know how to use $T$ and $T-I$ are invertible these two properties. Thus, any suggestions? Thanks!

Answer 1

Hint: Note that $$ T^2 = T^* \implies (T^2)^* = T^{**}\implies (T^*)^2 = T \implies (T^2)^2 = T. $$ Use the invertibility of $T$ and $T-I$ to rewrite the above equation as $T^2 = -T-I$.