New to the Einstein notation, having trouble with basic calculations

[Ineedhelpimbadatphys]

Homework Statement

picture below.

Relevant Equations

no equations, but I am most interested in c), d), g) and e). I would like to know if my attempts are correct. If not, what am I doing wrong?
We are using minkowski metric.

 

[Ibix]

I haven't checked your arithmetic, but (c) seems correct, if long-winded. You could just say $A^{(ab)}=\frac 12(A^{ab}+A^{ba})$ by definition.

Similarly (d), where you can just say $A^{[ab]}=\frac 12(A^{ab}-A^{ba})$ by definition. I suspect going the long way round got you into a pickle here, because you've somehow ended up with extra factors of the metric that shouldn't be there. (By the way, don't put commas between indices. Some people use $V_{a,b}$ as shorthand for $\frac{\partial}{\partial x_b}V_a$, and you're liable to be misinterpreted.)

Your approach to (g) and (h) (did you mean (h) or (e)?) appears correct, but I'm not sure you've lowered indices correctly on the tensor. What are the components of $\eta_{ab}$ in your convention?