You are right. The BCCI is very supportive of Virat Kohli. And that is what should be done. Because Virat has performed the duties of captain so diligently so far. Now he is in somewhat off form. Hopefully, in a few days, he will find out his mistakes. Take appropriate steps to correct his mistakes. And go back to the previous destructive form again
The problem with Virat Right now is that his aggression has got converted to arrogance and ego. As a result, he is unable to take unbiased decisions and in the end it affects the winning probability of the team. A lot of people have already pointed out his mistakes. But rather than rectifying them, he has attacked those who pointed out these facts. I don't think that he will ever change, unless the BCCI takes some concrete action, such as making Rahane as the test team captain.
Virat Kohli is naturally a bit arrogant. And now he has become somewhat aggressive. If he makes any important decision on the field, he can discuss it with Rohit Sharma and Rahane. This reduces the chances of making a mistake. He must find fault with himself, and strengthen himself in those places.