"Can I consume only coconut oil and water for a few days to start ketosis?"
While you can consume only coconut oil and water for a few days, to go into ketosis, however unless there are any actual medical reasons for going on a keto based diet, there's no real reason to do so. The reason that's what's known as ketogenic diet was for those who suffered from seizures, for controlling them and then fell to the wayside, as medications that controlled the seizures came out. However it's also been found to have a positive effect on those who has a sugar problem. The ketogenic diet is also not meant to be a long term diet, though some has tried it as a weight loss method, and to some degree used it as a short term method for insulin resistance. It's also a diet that requires a fairly good understanding of nutrition, and without that, the guidance of a registered dietitian, who is familiar with the ketogenic diet, should be consulted.