I'm having trouble proving by induction that this following Fibonacci-Lucas equation

$$F_{2n+k} = F_n L_{n+k} + (-1)^n F_k \tag{*}$$

is true, given that

$$F_{2n} = F_nL_n$$

and

$$F_{2n+1} = F_nL_{n+1} + (-1)^n$$

are true.

I did the base case $k = 1$, but I can't prove the induction step for $k+1$. In particular, my textbook said I have to assume (*) is true for $k$ and prove it for $k+1$, but I cannot prove it without assuming (*) is true for $k$ AND $k-1$.

Can someone help me? This is the first time I'm posting so I'm sorry if there's anything wrong.