I need to classify all elements of order 3 up to conjugation in $PGL(4,\mathbb{R})$. It's sufficient to give a representative of each conjugacy class.
My thoughts: consider instead $GL(4,\mathbb{C})$ where these elements are diagonalizable. But then I should find a matrix in the same conjugacy class with real coefficients and I don't see how can I do that.
Thanks!
I don't see that there is any advantage working in ${\rm PGL}(4,\mathbb{R})$ here, rather than ${\rm GL}(4,\mathbb{R}).$ If $x^{3}$ is a real scalar matrix, then a real scalar multiple of $x$ has order $3,$ so we might as well assume that $x$ has order $3.$ If you want to be really pedantic, then $\lambda x$ and $x$ are never conjugate in ${\rm GL}(4,\mathbb{R})$ for $\lambda \neq 1$ a non-zero real number, and $x$ genuinely of order $3.$ Now for $x$ genuinely of order $3,$ you can easily see that $x$ has trace $1$ or $-2.$ I won't finish the argument, but there are just the two possible conjugacy classes of elements genuinely of order $3$ in ${\rm GL}(4,\mathbb{R}).$