The problem you have to deal with is that all ways to connect a 4-pin fan to some power source assumes that the connection on the FAN side is a standard one for ALL fans. That is NOT true for this fan. So there is BOTH a mechanical fit problem and an electrical signal match problem.
IF you want to rig up your own electrical connection, you can experiment to find the right way. With no documents, we cannot know which wire is which on the connectors. By doing this you can find out which two are for supplying power to the fan motor. The other two will be associated with control of the lights. You have no good way to supply power to the lights or control them, so you can just forget about those entirely. The process for identifying the motor leads MIGHT risk damaging the lights, but you can't use them anyway.
You need a couple of wires and a battery. Although the fan will need a 12 VDC supply, it should be able to start up and run slowly with 6 VDC. So try to arrange such a supply from a 6 V lantern battery, or four common D cells connected in series. Ensure you know which wire is from the + end of the battery. Put the - wire into one hole of the fan's female plug, I suggest start with one on an end. Then, one hole at a time, push the + wire into another hole briefly and watch whether the motor turns. If not, pull the wire and go on to the next one until you've tried all three. If none of that works, reverse the wires: put the + one into the first hole and try the - one on each of the other three.
If that does not get the motor to start, try the next sequence. - one into the second hole, and try the + on the 3rd and 4th. Do not bother trying + in the first hole - you've already done that in the first sequence. If no success, reverse by putting + into second hole, and - to the other two. If still no success, last two choices are the wires on holes 3 and 4, in each direction. Somewhere among there you will get the motor to spin. MARK those two holes with + and -.
Just to do a final check: that should only have made the motor spin in ONE set of connections - all the others should have failed. You MIGHT even have made a bit of light flickers, but that does not matter. But if you got the motor to spin in more than one case, there's a problem still with knowing which wires to connect!
Now you know the fan motor power lines. You need then to find a way to rig a connection from your mobo's only CHA_FAN header to the right holes on the fan connector. The easiest way would be to get an old fan that is junk and simply cut off its wire cable at the fan so you can connect that to your fan's connector.
Connection details we know for a standard fan header on the mobo. It has four pins sticking up, and beside them is a plastic tongue sticking up beside pins 1 to 3. Pin #4 is past the tongue; Pin #1 is at the other end. Pin #1 at the end by the tongue is the - or Ground connection. Pin #2 in the middle of the tongue is the +12 VDC connection. So trace the wires on your recovered set from the old fan, and arrange to connect from the female connector on its end (and matching the pins 1 (- Ground) and 2 (+12V) on the header) to the holes you marked on the female connector of your new fan in your search above. When you have that done and secure, as a proof check connect the new fan with its modified connection system to your mobo's CHA_FAN header. Make sure the fan is secure and not about to hit anything, and turn on your computer. The fan should start up and run, VERY likely at constant full speed, right away. You can shut down now and mount the fan, or you can do the next step first.
The new fan is designed as the older 3-pin fan type, but you will be using only the two power pins in your connection. To get control of the fan speed, you need to make a small adjustment in BIOS Setup. See your mobo manual. If you have not gone into BIOS Setup before, see p. 34. If you get to EZ Mode (p. 36), click on Advanced Mode at upper right.
See p. 38 and click on HW Monitor at top middle. See p. 62 and set the options as:
Control Mode to DC
Setting to Standard
Temp Source to mobo, not CPU
When done, use ESC to get back to Advanced Menu, then F10 to get to the Exit Screen (p. 67). There choose Save Changes and Exit to save your settings and reboot. Now you should see your fan start up full speed for just a few seconds, then slow down a lot because your system is still cool. NOTE that, if you get an error message on screen that the CHA_FAN has failed, you can ignore that. I don't see a place in BIOS Setup for you to shut that off entirely. The fan header monitors the speed signal from the fan and may send you a message if it does not get one. But your fan appears not to have a speed signal wire, so you have not connected one for the header to "see".
If you have not done so already, shut down and mount your fan, then ensure it is plugged in, and close up the case. You're done!
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