Is there a name for this Fibonacci Identity

Vajda's identity states that (I have changed subscript variables to reassign then to derive your equality): $$ \left|F_{r+s}F_{r+t}-F_rF_{r+s+t}\right|=\left|F_sF_t\right|. $$

The above identity can be derived from this by setting $r=i-k$, $s=k+j$, and $t=k-j$.

This is equivalent to an identity of Alberto Tagiuri: equation (3) in Di alcune successioni ricorrenti a termini interi e positivi, Periodico di Matematica 16 (1900–1901), 1–12.