This is a review article that discusses an action of insulin that is not normally considered, it's possible function as a vasodilator. I wonder if this might not be the reason why some low carbers still struggle with high blood pressure even after achieving a lower weight.
... insulin-induced glucose uptake was associated with a striking increase in leg blood flow; the insulin dose-responses of blood flow and glucose extraction were very similar. These findings raised the possibility that insulin is a vasodilator hormone. In addition, the effect of insulin on leg blood flow was blunted in obese insulin-resistant patients, suggesting that vascular (in)sensitivity is a component of insulin resistance.
My summary: There appears to be a vasodilating action of insulin, however it is unclear whether this is a direct action of insulin or through an indirect action due to insulin's influence on lipolysis. Insulin inhibits lipolysis (not "fat burning" but merely the conversion of triglycerides to free fatty acids), so lower levels increase NEFA/FFA's