New to the Einstein notation, having trouble with basic calculations

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    Einstein notation
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The discussion addresses challenges with Einstein notation and basic tensor calculations. It suggests simplifying expressions for symmetric and antisymmetric tensors using definitions, specifically highlighting the correct forms for A^{(ab)} and A^{[ab]}. The conversation points out potential confusion stemming from unnecessary factors of the metric and advises against using commas between indices to avoid misinterpretation. Additionally, it questions the accuracy of index lowering in the user's calculations and prompts clarification on the components of the metric tensor. Understanding these nuances is crucial for mastering Einstein notation.
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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.
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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?
 
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