The prolific green algae Chaetomorpha spp. has become one of the most widely used “macros” by saltwater aquarists. Generally, it is maintained in large clumps within a specially set up refugium. In some other cases, it is kept in so-called algae reactors (indeed, Chaetomorpha is by far the most commonly used algae for this). However, once in a while, a hobbyist inquires as to whether or not it is possible to cultivate these algae inside the main tank alongside corals, clams, etc.
The short answer is, yes. Of course, you can put it in there. The better question, however, is “do you really want to?” And “if so, why?”
"Back in the day,” a lot of hobbyists similarly experimented with Caulerpa spp. Explaining that they liked the “greenery” and just wanted to have a little bit for variety, they’d throw a small wad somewhere on the bottom of the display. The problem with doing this is that the caulerpa could grow faster than most hobbyists would prefer, resulting in frequent trimmings. Without constant trimmings and pruning, chaeto can grow over corals and sessile inverts.
So, maybe “chaeto” isn’t quite as invasive as Caulerpa. But the same sort of thing can occur. Chaeto is messy and can grow quite rapidly under the “right” conditions. Just saying...
If you really, really like pruning and picking at your tank, then, by all means, add some chaeto to the display. But we could only suggest such a thing--all the above cautions thoroughly emphasized--if you intend to maintain small, ornamental patches of the stuff. Also, it is generally accepted that constantly putting your hands in the aquarium is a bad thing. Your hands of oils on them that can disturb corals. It also increases the chances for you to knock something over. Saying, "I'll be really careful to not knock stuff over" is easier said than done especially if you are putting your hands in the tank daily. Another thing, chaeto is the messiest macroalgae. With the frequent clippings, your tank is almost guaranteed to be littered with pieces of chaeto "noodles". It will get frustrating fast.
If you are determined to keep macroalgae in your display, that is awesome. Consider keeping a slower growing macroalgae such as Blue Hypnea or Dragon's Tongue. These macroalga species are slower growing and also more visually appealing. Most red macroalga are going to be slower growing than green.
If you are determined to showoff chaeto in a display aquarium, consider having a macroalgae specific tank, a sort of planted saltwater aquarium. However, keep in mind the chaeto will still need to be trimmed to keep it from choking itself out.
If you intend to keep enough chaeto in your system to reliably maintain low dissolved nutrient levels, then we must suggest that you grow it in a refugium or reactor only! It is simply what chaeto is best at. Feel free to try putting some in your display. You will quickly realize why it should only be kept in a reactor or refugium. It is still a great macroalgae for nutrient export and as a copepods habitat, just not great for the display.