Part a) of this is fine, but I'm really stuck on part b) and I have a test on this in an hours time, does anyone have any hints?
closed as off-topic by Toby Mak, Dietrich Burde, воитель, mrtaurho, Yanior Weg Aug 12 at 19:32
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I think the best place to start is to first give things names, e.g.
$$(a,b,c) \ast (\lambda, \mu, \nu) =: (x,y,z) $$
And then simplify the RHS of the expression
$$ F(x,y,z) = F(a,b,c) \cdot F(\lambda, \mu, \nu)$$
to get an expression for $x$, $y$ and $z$. You'll use part a) for this.
EDIT: $(\alpha, \beta, \gamma)$ were not the best choice of letters!
Then use the relation $\alpha^3=-3\alpha+24/5$ to reduce this to something like
which will give us a way to compose $(a,b,c)*(d,e,f)=(-3ad+af+be+cd,24ad/5-3bd-3ae+bf+ce,24bd/5+24ae/5+cf)$