Some generic advice (no elise never mind the LSS wheels)
1) The wheels are clear anodized, I hear. If true then be VERY careful about using cheapo wheels cleaners. You can stain the surface in one application. Something like soapy water alone might be adequate. If not, then the P21S stuff is nice and mild. I'd confirm what the finish is on the wheels before I used more than soap and water. The stock wheels look like they are powder coated hence they should be somewhat more durable WRT cleaning agents.
2) Nook and crannies - I'm used to them from BBS wheels on my Bimmers. Get a suitable brush with decent bristles. Some of the detailing places have really good ones. Griot's for example. You can also uses brushes you make/modify from painting brushes with snipped bristles.
I agree with the previous comments (especially the P21S recommendation) and would add that waxing the wheels may help them clean up easier. In addition, it may provide some more protection from stone chips, etc.
Tire Rack sells a wheel wax although I know others use normal car wax on the wheels.
A suitable wheel brush can really help if it fits where it needs to fit. Here's an example from Griot's:
Griot's also sells natural fiber brushes that are supposed to be amazing for car washes. I'd imagine that a certain kind of gentle brush might be nice to have for cleaning the various grills and contours of the Elise. The Elise has a LOT of grills!
One of our local Porsche dealers in Dallas uses a product called Reje-X to protect paint, glass and virtually anything else from birds, bugs, stains, etc. It was developed for use on airplanes and trucks and busses to make a very slick surface that bugs and exhaust gasses won't stick to. I have used it on my black 911, and dust and bird droppings vanish with little effort other than a stream of water from a hose. I suggest that it might be good for the diffuser to prevent exhaust soot stains (it's primary application) and on the LSS rims to prevent brake dust and grime from adhering. Detailing purists will object that it is a surface-bonding polymer and not a carnuba wax, but in my experience carnuba looks nice, but "evaporates" quickly in Texas heat.
My advice is to not use any wheel cleaners at all! I use a polymer sealent on my wheels, specifically Klasse All in One followed by 3 -4 coats of Klasse Sealent Glaze. This will last about 6 months and your wheels will clean up with whatever you use to wash your car. Believe me your wheels will be very easy to clean.
I Zaino'd my G35 wheels as an afterthought ("What to do with the leftover activated Zaino?") and it's turned out to be as cool as the backside of the pillow.
Brake dust just rinses right off, and the (clearcoated) wheels stay shiny. Much less stuff 'embeds' into them.
Now I make sure to keep enough Zaino in the applicator to get the wheels each time. I expect the Elise wheels to be the same.
Still doesn't protect from curb rub though.. The G35 spokes protrude further than the rubber, making parallel parking really tricky. How are the Lotus wheels? Same deal -- metal extending beyond the profile of the tire?
definitely Rejex. It will keep the brake dust from sticking. A guy on my Porsche board just started www.rejexusa.com . He is a good guy so if you don't have a reason to buy it somewhere else, you can feel good about supporting his new venture.
Weekly washings will be a snap with regular soap and water. Othewise follow other's advice.
If you live near a motorcycle shop, go down and see if they carry S100 brand products. Pick up either the wheel cleaner or total cycle wash. Follow directions and that should save you some time.
Guru reports seems to be recommending Eagle One A2Z which is readily available anywhere, but I suspect that you cannot let it sit for very long at all. So this product, though I haven't used the latest version, requires a bit more care compared to the S100. Griots also makes a good wheel cleaner.
Carnauba is okay for coating your wheels, but heat and wheel cleaner will strip it right off.
I'm told the boar's hair brushes require a bit of care, especially with softer German clear coats. You should be very careful when using these, but generally they work. They are not good with heavy dirt, but neither is sheepskin. Personally, I'm sticking to sheepskin.
I'm chiming in with my vote for P21S/S100 wheel cleaner. If they are anodized, any acid wheel cleaner will etch them (ask any Porsche owner about his Fuche wheels). I would be equally uncomfortable using a strong base such as Eagle One A2Z or even einszett's wheel cleaner. Although that is what I used on the BBS wheel in the picture.
Like others said, a good synthetic such as Klasse will make the cleaning process much easier. Twice a year I pull my wheels off and give them a super cleaning. I finish them with a coat of AIO and SG.