I have the following projectivity: $$ f[x_1,x_2,x_3]=[4x_1+2x_2-x_3,2x_2,x_3,-x_2-x_3]. $$
I have to find all the lines $L$ such that $f(L) \subset L$.
I've found the eigenvalues of this matrix, which are three distinct real values. So the lines that satisfy the above conditions are only the fixed lines passing through two of these points or there are many other? Does a line passing through only one of these points satisfy the request?
The projective lines invariant under $f$ are images of $2$-dimensional subspaces invariant under $f$.
Since the eigenvalues are distinct the only invariant subspaces of $V$ are sums of some of the $V_i$, so for $\dim=2$ you have $3$ such $2$-dimensional invariant subspaces.
Or you can think geometrically: $f$ as a map of the projective plane has $3$ fixed points. A line through any two of the fixed points is invariant ( as a subsets -- its points will be still moved around) since a line is taken to a line so the line through $2$ invariant points is invariant. Any line invariant under $f$ intersects another invariant line in a point that is fixed. It's clear there are only $3$ such invariant lines.
If the linear map $f$ had an eigenvalue with multiplicity $2$ that would give a line in the projective plane with all points fixed under $f$.