There's a bit to unwrap here.

On a low carb diet or fasting, there's a net release of free fatty acids into the circulation from the fat cells, leading to an increase in blood free fatty acids, this is where much of the fuel for your body comes from. The liver does produce ketones from these free fatty acids, as a result of a lack of glucose. Most cells in the body can use these ketones, but further into the diet, they'll mostly be spared for the brain (increased adaptation to fat for energy causes the muscle cells to convert acetoacetate into betahydroxybutyrate, this effectively saves it for the brain, since it's best adapted for the conversion back to the acetoacetate that's a more direct fuel source).

The liver does repackage free fatty acids as triglyceride for export as VLDL, these can be taken up by other cells to increase fat uptake to higher levels than can be supplied by the free fatty acids in the blood, when needed.

The ketones are a product of the liver producing energy from fat. The link between ketones and triglyceride is the high levels of free fatty acid in the fasted or ketogenic state, rather than the one being the product of the other.

When it comes to muscle mass--my Dad is 73. In recent years, on his low carb diet, he has been losing lean mass, we put it largely down to age since he did get some exercise working in casual construction jobs. But he's changed a few things, and that's reversed.

He started eating all meat, and higher protein. Also he's started driving a school bus--and having a sedentary job now, decided to join a gym and get more structured exercise. I think it's probably like 90 percent exercise, 10 percent diet, because his protein intake was really never very low. I guess I'm saying, do you even lift, bruh?