Finding the transition matrix for the rational canonical form

Finding the transition matrix for the rational canonical form

Let $A$ be the 3x3 matrix
$$\begin{bmatrix} 3 & 4 & 0 \\-1 & -3 & -2 \\ 1 & 2 & 1 \end{bmatrix}$$
The characterisitc and minimal polynomials are both $(x-1)^2(x+1)$
The eigenspace for 1 is $$\{ \begin{bmatrix} 2 \\-1 \\ 1 \end{bmatrix} \}$$
The eigenspace for -1 is: $$\{ \begin{bmatrix} 2 \\-2 \\ 1 \end{bmatrix} \}$$
The rational canonical form $R$ is:
$$\begin{bmatrix} -1 & 0 & 0 \\0 & 0 & -1 \\ 0 & 1 & 2 \end{bmatrix}$$
I want to find the transition matrix $P$ such that $A=PRP^{-1}$
I thought we had to find 3 independent vectors...one from the eignspace of
1, another from the eigenspace of -1, and then any other third vector such
that the three would be linearly independent. So I chose P to be:
$$\begin{bmatrix} 2 & 2 & 1 \\-2 & -1 & 0 \\ 1 & 1 & 0 \end{bmatrix}$$
But when I multiplied $PRP^{-1}$, I did not get $A$...I'm not sure why.
I would appreciate it if anybody could tell me where I went wrong and how
I can fix it.
Thanks in advance