Linear Alegbra - Is this linear transformation isomorphism?

Question

Let $\mathbb R^4 \rightarrow \mathbb R^4$ linear transformation.

That : $$\dim\operatorname{Im}(T+I)=\dim\ker(3I-T)=2$$

Is $T-I$ isomorphism?

The only thing I come up with is that 3 and -1 are eigenvalues, that has geometric multiplicity of 2, and alegbraic multiplicity of 2, so T diagonalizable.

But how should I find out if it isomorphism or not? Any ideas?