Consider the two matrices with some parameter $s \in \mathbb R$

$$A_1= \begin{pmatrix} s& -1 &0& 0 \\1&0 &0&0 \\ 0&0&1&0 \\0&0&0&1 \end{pmatrix}$$ and $$A_2= \begin{pmatrix} s& -1 &-1& 0 \\1&0 &0&0 \\ -1&0&s&-1 \\0&0&1&0 \end{pmatrix}.$$

I then noticed that the eigenvalues of arbitrary products of $A_1$ and $A_2$, i.e. e.g. $A_1A_2A_1$ and $A_1A_1A_2A_1$ etc. all have eigenvalues $\lambda_1,1/\lambda_1$ and $\lambda_2, 1/\lambda_2.$

It is clear that the product of eigenvalues is equal to one, as both matrices are in $\text{SL}(4,\mathbb R)$, but I don't see why they have to come in two pairs that multiply up to one, respectively.