lp4 is also known as Molex 4-pin, a connector design on power supply lines coming out of your PSU. Originally used for power to IDE devices like HDD's and optical drives. Many PSU's have several of these, but some older ones did not have very many added power connectors for SATA units. So case manufacturers often include a few adapters to convert excess Molex 4-pin (aka lp4) power connectors to SATA power connectors in case you need them.
Not sure what you mean by "tricool adapter...". Antec Tricool fans come with a 3-pin input connector on their wires, but many also have attached to that an adapter that lets you connect directly into a Molex 4-pin (aka lp4) power source. If you want to connect the fan directly to a mobo 3-pin SYS_FANx port, you remove the adapter.
Oh, if you have 4-pin fans, you will have to check carefully whether the fan controller you want to use can handle them. IF you have 4-pin fan headers on your mobo, those could control your fans.
4-pin fans are controlled by PWM signals, quite different from 3-pin. But the physical construction of the connectors makes it possible to plug one into the other for limited compatibility. A 3-pin system has lines for +12 VDC (red), Ground (black) and fan speed (yellow). Fan speed is a pulse signal (2 pulses per revolution) generated by the fan motor and sent back to the mobo (or controller) for monitoring. Fan speed is controlled by varying the +12 VDC value to less than 12 volts.
A 4-pin fan system uses those same 3 lines in the same physical layout (but uses different wire color codes), plus a 4th line that carries the PWM signal from controller to fan motor. The difference is that in PWN control mode, the +12 VDC signal never changes - it is always at 12 volts. The PWM signal is a square wave with varying "% on". Inside the fan motor is a tiny local controller that uses this PWM signal to control exactly when the motor actually receives current from the 12 volt source, and that's how speed is controlled.
If you plug a 3-pin fan into a 4-pin header, it will receive 12 volts all the time and run always at full speed. An exception to this is that some 4-pin headers are fed by controller circuits that can be changed to act as 3-pin controllers and vary the +12 VDC line.
If you plug a 4-pin fan into a 3-pin header, I am not really sure whether it works properly or not - maybe someone else can clarify. The fan will receive a +12VDC supply that is NOT constantly 12 volts, and it will receive NO PWM signal. In that case I am not sure whether the motor and its mini-controller can just run according to whatever voltage it receives.
That last question I can't answer is at the heart of whether you can use the fan controller you want with 4-pin fans. IF the controller is designed for 4-pins anyway, no problem! But if it is designed for 3-pin fans, you need to know whether the 4-pin fans you have can work with that type of supply signals.
OK, your picture makes this clearer. These are NOT what is called a "4-pin fan". If anything they might be called "2-pin fans". Note that each fan has only 2 wires to it - black is ground, red is the + VDC line. These fans typically run at full speed at 12 VDC, and need at least 5 VDC to start up. They have NO third (yellow) lead to feed a fan speed signal back out. They do not have a fourth lead for PMW signal. The "4-pin" connector you see on the ends of the leads is just a 4-pin Molex (aka lp4) connector that plugs into one power supply output. They may be 2-sided so that you can stack two or three and plug them all into one Molex PSU connector. As jsc points out, Antec usually includes with their Tricool fan a small 3-position switch that also connects to it so you can choose what speed it runs - Low / Med / Hi kind of choice. These fans do NOT plug into a 4-pin or 3-pin mobo SYS_FANx header to allow the mobo to control their speed. They plug into a Molex 4-pin connector and you manually fix their speed with Antec's switch.
Well, if you want to connect them to a 3-pin header output on you mobo, you most certainly can do that either using an adapter or a by installing a replacement connector on the end of the wires. You'll just need to pay attention to four things.
1. Check the wiring of the mobo pinout. Make sure to connect the RED wire to the +12 VDC pin, and the BLACK wire to the Ground pin.
2. Done this way, your fan will not send any speed signal back to the mobo. Check in your BIOS setup screen for the fan output you're using - I assume it will be one of the SYS_FANx ports. If there is an option , set it to Ignore the fan speed (there is none) so the mobo won't send you alarms the the fan is not turning.
3. If the fans have their Antec TriCool 3-speed switch permanently attached, set it to full speed so it does not reduce the voltage - the mobo will do that to control fan speed.
4. If you are re-wiring the fans, you can take the option to wire two fans in parallel from one mobo output. Then they will both change speeds according to the mobo's speed control system. I would not try wiring 3 in parallel, though - not sure the port could handle the start-up current for 3.