You can literally just do an atomic swap on some memory location. Three lines of assembly, like.

    mov rax, 1  ; load the value to exchange into rax
   acquire_lock:
    ; attempt to acquire the lock
    xchg byte [ADDR_LOCK], al  ; atomically swap the lock value with rax
    test al, al  ; test if the original lock value was 0 (unlocked)
    jnz acquire_lock  ; if it was not, loop until we can acquire the lock

The downside is you want a backoff to sleep the thread so it doesn't go into a loop. But the actual lock code is simple. You can easily have this be your function "AcquireLock()" and then do

while(!AcquireLock()) { //pause thread execution. }

And I think this is where they get the syscall being needed, since this will normally require a syscall to pause the thread from the scheduler.