There are several "sixteen Lancers" quadrilles, written for eight couples rather than four, many of which adopt similar solutions in adapting the figures to twice as many couples. The Royal Lancers, or Horse Guards, is a version that I have found in only two sources: Thomas Hillgrove's 1863 A Complete Practical Guide to the Art of Dancing and Professor M.J. Koncen's 1883 Quadrille Call Book and Ball Room Guide. The diagram at the left (click to enlarge), taken from Hillgrove, shows the arrangement of couples for the Royal Lancers. Note that the odd-numbered couples on each side are to the right of the even-numbered ones. Koncen provides a similar diagram but misnumbers two of the couples.
The figures of the Royal Lancers are only minor variations on the standard Lancers ones, but the multiplier effect of doing the figures with twice as many couples makes even the simplest figures substantially more interesting. For the most part, the Royal Lancers may be danced to any Lancers music, but note that there are both 20-bar and 24-bar versions of Figure 4 in the standard Lancers repertoire, and the Royal Lancers requires the longer music.