Install DDR3 1333 RAM in DDR3 2200 Motherboard Compatibility

Yes. As long as both modules are supported by the board, the faster will clock down to match the slower. Enjoy your 8GB of ram.

Also, you don't have to worry too much about breaking your motherboard as long as the memory types matches the slot. At worst it'll fail to POST. Google 'Power On Self Test' for more.

TL;DR Yes, you can.

What's given is:

1. A motherboard that supports DDR4 speeds up to 3200MHz.
2. A CPU that supports DDR4 speeds up to 2933MHz.
3. And you that want to install a RAM module that is 2666MHz.

Before we get the answer let's consider a few examples.

First example: We have a motherboard, that's maximum supported RAM speed is 2666MHz, and a CPU, that's maximum supported RAM speed is 2966MHz. What's gonna happen if we install a 3200MHz RAM module? The answer is motherboard will force RAM to operate at motherboard's maximum speed which is 2666MHz.

Second example: We have a motherboard that's maximum RAM speed is 3200MHz, CPU's maximum is 2400MHz. What's gonna happen if we install a ram module with crazy 3600MHz clock? The answer is motherboard will drop down RAM speed to match CPU's maximum supported clock speed which is 2400MHz. Even though you spend a couple hundreds of dollars on 3600MHz DDR4 module you will still only get 2400MHz from it, because CPU is gonna be a bottleneck for it.

And for better understanding of how all this throttling thing works, the third example: Our motherboard and CPU can work with 3200MHz memory. The first RAM module's speed is 3200, the second's is 2933, and the third one's is 2133. Whose clock speed will motherboard select? The answer is the lowest of all memory modules, which is 2133MHz. Every single module will be operating at the lowest speed of all of them. 3200, 2933 and 2133 will throttle down to 2133.

Now we can make a conclusion: Motherboard will throttle down RAM clock to the maximum CPU's supported RAM speed and/or to the lowest of all installed RAM modules.

So yes, you can install a 2666MHz module on this system. Any module that's lower than 2933MHz will be perfectly fine, even 1600MHz. But the best choice would still be 2933MHz module.