Kid not doing homework

[Sparklette]

Has he been tested for any learning disabilities? 
He could also be twice exceptional (meaning he is gifted but also struggles from learning disabilities.) 
All students are eligible to be tested for learning disabilities by their public school (or if in private school, by the public school district that the private school resides in.) 

Is the homework that he isn't doing at home, additional assignments to what they do in school, or is the homework items he is not completing in class? If it's items he's not completing in class, then you need to find out why he is not able to complete them if the other students are. (If it takes a student longer to complete the same assignments as the average student, that could point to a learning disorder or processing disorder, which would make him eligible for accommodations, such as 1.5 or 2 times the time for taking tests or for writing assignments in class, etc. Also, documenting such academic needs of the student is important if they take the path to college and need accommodations for standardized tests and college courses, etc.

You might want to check out http://www.parentcenterhub.org ;  
https://sites.ed.gov/idea/ ; 
https://www.greatschools.org/gk/articles/section-504-2/ 

There is also the possibility of a 504 plan that can be utilized if he needs only accommodations (such as extended time or testing in distraction free areas, etc)

Just my 2 cents as I don't believe anybody has mentioned this possibility in the prior responses, and I'm well aware of how such accommodations and proper supports changed the trajectory of high school for my foster/adopt child who has blown everyone's expectations of academic achievements out of the water. 
(My FAC graduated a year ahead of high school cohort class as a Junior at 16 with over 30 college credits (all from community college courses both F2F and online) followed by an associate degree the follow year a month ahead of cohort HS class graduation from HS. No thanks to any of the public schools FAC attended. Had to get private tutoring after losing battles with the public school district to provide proper remedial services/instruction; but well worth the out of pocket expenses as it just shows that breaking the cycle of poverty and educational struggles of Title 1 and Students with Disabilities is actually possible if the proper instruction is provided. And its never too late, as FAC was 15 and finished up while attending college at 17;  although earlier is definitely better, as then they don't burn in neural pathways that will need to be rewired later on!)

I was in the gifted program back in elementary and junior high and don’t really see him as on par with the kids I knew there. He seems to “get” math but has no follow through and makes a bunch of mistakes until his dad checks his work. Writing is not his thing. I think the problem there is he won’t read anything. He only seems to learn about a topic if it’s to show his older brother up. That at least demonstrates he’s capable of learning, but needs motivation. He does seem to have an intuitive knowledge that it’s not useful or necessary to know about a lot of the things that are being presented, but hasn’t accepted he needs to do the assignments to get through the course.
I have him and the older brother doing supplemental reading projects with the hopes of him finding an area of interest. I marginally think it’s military history, but only some days. I’ve tried giving him physical projects to build, but he just races through with no real enjoyment. Doesn’t like sports. Loves video games if he can do well in them. Really likes to be superior at things and dismisses anything he is not excelling at as unimportant. Does seem to have a natural grasp at negotiating, but only when the goal is for his own benefit.