Sometimes the obvious needs to be said as the answer is that simply. Some things in life cannot be explained by reductive material science. Yet are just as real as the objective world.
So we find it hard to explain it in material terms. But we understand and believe its real within ourselves. Perhaps you could say intuition, sixth sense, or phenomenal knowledge. Whatever its hard to explain and cannot be evidenced by objective science yet is real.
You are supporting something and you have no apparent idea of how it works.
No we do have some idea, its one of those things, like conscious experience of red. Try explaining that to someone. You canot in material terms. But you know within yourself that its real, its a real phenomena, a red experience just as much as the chair we are sitting on.
Its almost like we have the same level of belief as sensing something is real objectively, seeing and touching an object but without an object. Its more transcedent phenomena but every bit as real that we believe so like its sitting there in front of us.
You must have heard of the Hard Problem of consciousness. Basically its the problem that we know consciousness is real but we cannot explain its nature. The same with free will. It falls into the conscious experiences, real but hard to explain as they can only be knownm by the experiencer, the subjects experience of free will.
I have given you quite specific answers as to how one comes to exercise one's 'free will', how one comes to make choices. How they are determined. And you've offered nothing in return. Except variations on 'It seems obvious that we have it'.
Like consciousness, its obvious to us but we cannot put it in the terms you want. Of course I know all the material reasons why we have no free will. But that comes under the reductionism and physical processes. Ones we can see, hear and test mechanistically.
But there is another layer of reality that has influence in the world. We know that because we live by it everyday. The transcedental aspects of reality like belief, experience, agency because they are of the mind and subjective experience. The materialist claims this is all an epiphenomena. Its not anything real in the world itself, Diesn't fundementally influence reality let alone the physical.
So I can't give you any material examples just like I can't give you an 'ought' from an 'is' or a red experience from a neuron and light wave. I have to step into the transcedental, the Mind over matter realm to be able to give examples. Like our proper basic belief that our free will is real.
A proper belief is not one that is unjustified but the result of our experiences, testing free will, seeing the results directly, knowing how this is connected to our very being in many different ways. Enough personal evidence to cause us to believe we have control in the world even to the point of believing and knowing that our belief is not delusional but based on reality.
We know how we make decisions. I can go into a lot of neurological details if you like (oh, the horror of materialism!). To save me the trouble, explain the process whereby we make decisions that are not determined by any previous cause.
Yes as I said I understand the mechanisms. Man we could go back to the atom. Its all about the neurons, chemicals, genetics. The physical causes that cause and cause and cause again in a chain of events within the causal closure of the physical.
But none of that accounts for the nature of free will as experienced by the subject in the world. Thats another whole different realm of stuff going on that all those mechanisms and causes cannot account for. So you can go on about those cause and effect processes all day. Heck you can know everything there is to know about how the brain works.
But its still not going to explain or account for subjective experiences of the world, or reality. You can't find the experience of free will in neurons or brain activity. Thats just physical stuff reacting. It has no mind or consciousness.
Of course the material reductionist will claim that the Mind, consciousness, agency is just an by product of the physical stuff. But then thats like appealing to a ghost in the machine. Somehow unconscious stuff made consciousness when in the right combinations. Thats still a God of the gaps arguement. Put together unconsciuous stuff and poof out comes consciousness.
Material science has not explained experiences like consciousness including free will and agency. It can't. Thats the Hard Problem it cannot get around. So no amount of you explaining the how the brain works is going to explain free will or no free will.