Is it possible to make a Turing Machine that accepts the language $\{\alpha\alpha\alpha | \alpha \in \{a, b\}^∗\}$?
Here is a particularly simple algorithm for the slightly simpler language $\{\alpha\alpha \mid \alpha \in \{a,b\}^*\}$; a similar approach works in your case.

The Turing machine starts by checking whether the input is empty (if so, accept) and whether it has even length (otherwise, reject). Next, we are going to find the midpoint of the input. The idea is to put a "chip" on each end of the input, and at each step, advance each chip toward the middle, until the two chips coincide. If done correctly, we can mark the middle of the input, which is the starting point of the second copy of $\alpha$. All that remains to be done is to check that the first half of the input is identical to the second half, which can be done using a simple iteration.