  We obtain closed-form expressions for all sums of the form \mbox{$\sum_{k =
1}^n {F_{mk}{}^4 }$} and \mbox{$\sum_{k = 1}^n {L_{mk}{}^4 }$} and their
alternating versions, where $F_i$ and $L_i$ denote Fibonacci and Lucas numbers
respectively. Our results complement those of Melham who studied the
alternating sums.
