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EDIT: I misquoted the book, and the answers here show it isn't fixable.

Let $A,B \in SL_2(R)$. This book by Brumfiel and Hilden has the following two facts in it:

  1. $Tr(A)Tr(B) = Tr(AB) + Tr(AB^{-1})$.
  2. The pair $A,B$ is uniquely determined, up to simultaneous conjugation, by the elements $Tr(A)$, $Tr(B)$, and $Tr(AB)$.

(I don't have the book with me, but I can provide the exact references once I do.) These are valid over any commutative ring $R$ EDIT: (condition on $R$ to be filled in later).

Taking $A^2 = 1$ and $A = B$, the first fact implies that $Tr(A) = \pm 2$, and then the second fact implies that $A$ is conjugate (and hence equal) to $\pm Id$. Since there is an embedding $S_4 \hookrightarrow SL_3(R)$ (which is described in the other answers), we can finish by noting that the images of the transpositions are not central in $SL_3(R)$.

I would guess that there's a more elementary way to prove that involutions in $SL_2(R)$ are central, but I don't know one at the moment.
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Let $A,B \in SL_2(R)$. This book by Brumfiel and Hilden has the following two facts in it:

  1. $Tr(A)Tr(B) = Tr(AB) + Tr(AB^{-1})$.
  2. The pair $A,B$ is uniquely determined, up to simultaneous conjugation, by the elements $Tr(A)$, $Tr(B)$, and $Tr(AB)$.

(I don't have the book with me, but I can provide the exact references once I do.) These are valid over any commutative ring $R$. R$ EDIT: (condition on $R$ to be filled in later). Taking $A^2 = 1$ and $A = B$, the first fact implies that $Tr(A) = \pm 2$, and then the second fact implies that $A$ is conjugate (and hence equal) to $\pm Id$. Since there is an embedding $S_4 \hookrightarrow SL_3(R)$ (which is described in the other answers), we can finish by noting that the images of the transpositions are not central in $SL_3(R)$.

I would guess that there's a more elementary way to prove that involutions in $SL_2(R)$ are central, but I don't know one at the moment.
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Let $A,B \in SL_2(R)$. This book by Brumfiel and Hilden has the following two facts in it:

  1. $Tr(A)Tr(B) = Tr(AB) + Tr(AB^{-1})$.
  2. The pair $A,B$ is uniquely determined, up to simultaneous conjugation, by the elements $Tr(A)$, $Tr(B)$, and $Tr(AB)$.

(I don't have the book with me, but I can provide the exact references once I do.) These are valid over any commutative ring $R$. Taking $A^2 = 1$ and $A = B$, the first fact implies that $Tr(A) = \pm 2$, and then the second fact implies that $A$ is conjugate (and hence equal) to $\pm Id$. Since there is an embedding $S_4 \hookrightarrow SL_3(R)$ (which is described in the other answers), we can finish by noting that the images of the transpositions are not central in $SL_3(R)$.

I would guess that there's a more elementary way to prove that involutions in $SL_2(R)$ are central, but I don't know one at the moment.