I just encountered a very curious relation in an algebra. A bit simplified, the relation states that for expressions $a,b,c,x,y,z$ in the free algebra $\mathbb{Q}[X]$ over some (non-commutative!) alphabet $X$, we have $$ \begin{vmatrix} 1 & 1 & 1 \\ a & b & c \\ x & y & z \end{vmatrix}=0. $$ That is, $bz+cz+ay = az+bx+cy$.

We can think of $a,...,z$ as words, and multiplication is simply concatenation. Has this type of relation been observed in some algebras before? Does it fit into some larger picture?

In my concrete case, this relation does not hold in general, but only for certain particular combinations of $a,b,c,x,y,z$, so examples where there are some restrictions on when the relation applies are also interesting to me.