I'll answer the splitting question first because that's pretty straightforward. The answer is a Passive Splitter.
Screwfix sells the Labgear range of passive two-way and 4-way splitters. Some have an outdoor housing (blue cover), others are bare metal. They're designed for indoor use.
The TV aerial signal has two parts to it: There's the signal strength (power), and the quality which is a measure of how free from noise it is. Of the two, quality is by far the most important. Adding power is easy. Any amplified splitter or booster will do that. But there is no such booster that will improve quality. The best any of us can hope for is to not lose too much quality when we pass the signal through some or other device. Aplified splitters reduce the quality. Passive splitters maintain it, but they lose a little strength.
For best results, use a passive splitter as close to the receiving TVs as possible. This is the opposite of what happens with a powered splitter. They're used as close to the aerial as possible.
Now to the trickier question of that dome aerial. I have a problem with it; well, actually several, but I'll deal with the big stuff first.
There's no specification; and I don't just mean that the retailers forgot to add the detail, it's the manufacturer's web site too. There's no detail other than the basic frequency range it picks up. That's about as useful as going to buy a car where the brochure tells you nothing more than it can be driven on motorways. Any aerial worth a damn has a comprehensive specification that lists things such as gain (before amplification is added), what the aerial is like at rejecting unwanted signals, and how directional it is. Some of those things don't apply to a circular aerial designed to pick up from a 360 degree radius, but the gain should be listed. It isn't, and they don't even list the gain of the booster amp which is usually the trick with indoor amplified aerials. For this one though, there's nothing. To me as a professional installer that's a red flag. It says that they have something to hide from anyone who knows what's what with aerials.
The second problem is the size. Aerials work by resonating in sympathy with the signals they're designed to collect. Where we want a lot of signal from an aerial before amplification, then our choices are to increase the amount of metal in the air (a bigger aerial), or to make it more directional, or to tune it to a very specific range of frequencies. The dome aerial is small, completely the opposite of directional, and wideband. What all this means is that it doesn't collect a lot of the signal that you're interested in receiving which means poor signal to noise ratio (= low quality signal); and it will pick up from other local transmitters that you may not be interested in receiving, which makes the noise problem even worse.
The signal that the aerial delivers to the amplifier is weak and full of noise. All the amp does is make that noisy poor-quality signal louder.
The third problem is a bit more technical. It's to do with something called polarisation. We have TV masts all over the country. The main ones cover most areas, but there are some places such as in dips/valleys or behind hills that are in a reception black spot. To fill in these areas we use relay transmitters. Inevitably though there's an overlap in the fringe areas where the two signals co-exist. You'd think that would be a good thing; twice the signal, right? Except it doesn't work like that. With the transmitters being at two different distances the signals interfere with each other. That's not good. To get around this problem, the signals from nearly all relay transmitters are vertically polarised. The main transmitter signals are almost always horizontally polarised. An aerial is rotated to match the transmitter. In this way, signals can overlap without too much of an issue.
Thinking about this dome aerial, there's nothing in the description that says it suits one or the other transmitter polarisations. That means that with no adjustment required, it picks up from either. The way that is done is to design the aerial with a polarisation half way between vertical and horizontal. The catch is that it greatly reduces the signal strength and quality, so on top of all the other compromises, the aerial runs with a handicap that it never fully matches your local transmitter.
All things considered -
and I haven't even touched on the quality of the ancillaries - these are not great aerials.
Where you're looking for something unobtrusive, try a Log Periodic aerial. They're not a high gain aerial, but they do work well. If your home is in a very strong signal area then a min Log Periodic might be enough.
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