This is from Spivak "Intro to Differential Geometry" Chapter 10 Exercise 26.
We are given G as SL(2,$\mathbb{R}$). P is the inclusion map from G to $\mathbb{R}^4$. $x,y,u,v$ are the coordinate functions $x^{11},x^{12},x^{21},x^{22}$, respectively, and dP is the matrix of 1-forms (dx$^{ij}$). In parts a) and b), I was able to show that $\mathfrak{g}$ is the set of all traceless matrices and was able to compute P$^{-1}$dP.
Part c) asks to show that the 3-form
v dx$\wedge$du$\wedge$dy - y dx$\wedge$du$\wedge$dv
is left invariant. I'm a bit confused on where to start. Should I do this directly? There's a proposition in the text that $\omega$ is left invariant if $\psi^*\omega$ is right invariant where $\psi$(A) = A$^{-1}$. Is that a better route?
In part (b) of the exercise you showed (or could have deduced) that $v\,dx-y\,du$, $v\,dy-y\,dv$, and $-u\,dx+x\,du$ give a basis for the left-invariant $1$-forms. What do you get when you wedge them together to get a left-invariant $3$-form? (Oh, I now read the comments and saw you'd already thought of this. Yes, it certainly suffices! Note that you already know that any two (nonzero) left-invariant $3$-forms must be constant multiples of one another.)