There is a identity, well-known among people that know this sort of things, that is called Hall's identity (or Wagner's identity): for all choices of $2\times 2$ matrices over a fixed field $A$, $B$, $C$, we have $$[[A,B]^2,C]=0.$$ Here $[X,Y]=XY-YX$ is the usual commutator.
Does this generalize in some way to other contexts?
N.B. It is a result of Shoda in characteristic zero and of Albert and Muckenhoupt for any field, that every matrix of trace zero is a commutator, so Hall's identity tells us that the square of a traceless $2\times 2$ matrix is central.
N.B. If $Q$ is a quaternion algebra over a field $k$, then there is a an extension $K/k$ such that the result of extending scalars $Q_K$ from $k$ to $K$ in $Q$ is isomorphic to the $K$-algebra $M_2(K)$ of $2\times 2$ matrices. It follows easily from this that Hall's identity holds also in all quaternion algebras over all fields.
One way to understand Hall's identity is the following: it says that the function mapping two $2\times 2$ matrices $A$, $B$ to $[A,B]^2$ takes values in the center of the matrix algebra $M_2(K)$. There is a whole theory built around this idea: that of central polynomials.
Kaplansky asked (he was great at asking interesting questions!) if there were central polynomials for all sizes of matrices, and this problem was solved in '72 by Formanek and Razmysov, who constructed central polynomials for all sizes $n>2$.
The book by Procesi on rings with polynomial identities is a good reference for this.