So I have a set P={ $p(\alpha,\beta,\gamma)=\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$ $|$ $\alpha,\beta,\gamma$ $\in R$}. I needed to show P is a Lie group, which I have done.
I need to parametrise P, and I was asked to show that it is $R^3$, and, to show that the group multiplication is:
$p(\alpha,\beta,\gamma).p(x,y,z) = p(\alpha+x,\beta+\alpha z + y, \gamma+z)$
So I multiplied the matrices $\pmatrix{1&\alpha&\beta\\0&1&\gamma\\0&0&1}$. $\pmatrix{1&x&y\\0&1&z\\0&0&1}$ = $\pmatrix{1&\alpha+x&\beta+\alpha z + y\\0&1&\gamma+z\\0&0&1}$.
My question is, how would I parametrise the group above?