How to construct Lorentz invariant quantities from a set of Lorentz tensors? This is all the information given in this question. I have no idea how to build these Loerntz invariants. How would I go about answering this? 

 A: A Lorentz invariant is what we call a Lorentz scalar. Scalars don't have any upper or lower indices. 
For example, the 4-momentum $p^\mu$ is a Lorentz vector because it transforms the way a vector $V^\mu$ transforms, i.e $p'^\mu=\Lambda^\mu_\nu p^\nu$. A Lorentz scalar on the other hand stays the same when performing a Lorentz transformation $m'=m$. Here $m$ can be thought of as the mass of some particle.
If you contract a vector with a vector you get a scalar, $p^\mu p_\mu=m^2$ where $m$ is the mass. This is a Lorentz invariant because $p^\mu$ transforms as a contravariant vector while $p_\mu$ transforms as a covariant vector $$p'^\mu p'_\mu = \Lambda^\mu_\nu p^\nu \Lambda^\sigma_\mu p_\sigma = p^\mu p_\mu$$ where I have used $\Lambda^\mu_\nu\Lambda^\sigma_\mu = \delta^\sigma_\nu$.
So you can think about all possible contractions you can make from those given tensors, $T^{\mu\nu}$ and $a^\mu$.
A: Just combine tensors in a way that all Lorentz indices are contracted e.g $a_\mu a^\mu$, $a_\mu a_\nu T^{\mu\nu}$, $T^{\mu\nu}T_{\mu\nu}$ etc.
