If we take $\alpha = \frac {1 + \sqrt 5} 2$ and $\beta = \frac {1 - \sqrt 5} 2$ and $F_n$ be the $n$-th Fibonacci number and $L_n$ be the companion Lucas number then,
$$ F_n = \frac {\alpha^n - \beta^n} {\alpha - \beta} \text{ and } L_n = \alpha^n + \beta^n \text{ for $n = 1, 2, \cdots$ }. \tag 1 $$
We also have
$$ F_{2n} = F_n L_n \text{ and } L_n^2 - 5F_n^2 = 4(-1)^n \tag 2 $$
This gives a relation between the $n$-th Fibonacci/Lucas numbers and $F_{2n}$.
If we replaced $2$ with $m$ in the LHS of Eqn. (2), are there known relations for $F_{mn}$ in terms of the $m$-th, $n$-th Fibonacci/Lucas numbers? Please provide reeferences.
At https://functions.wolfram.com/IntegerFunctions/Fibonacci/introductions/FibonacciLucasNumbers/ShowAll.html under Transformations: multiple arguments we find several such formulas, e.g., $$ F_{mn}=L_mF_{m(n-1)}-(-1)^mF_{m(n-2)} $$