Your procedure is not clear to me.
Note that the title for your subject is "Noloben snake".

If your were able to go on top of Esher's lab and drop the slate with the snake symbol and then link to the main entrance bubble, run to the front of the door and wait for the door to open, you did ok.

If the door will not open, there are two possible causes.

1- you did not get in front of the door fast enough.

2- The Bahro did not like your drawing and you will have to replay the snake drawing (with your saved game) until the Bahro accepts your drawing. (or possibly a drawing done by one of your family's member or by a friend)

If you were able to get into the cave and see the Keep symbol (big circle + 4 small ones), you were supposed to link back to the main entrance bubble, get the slate brought back by the Bahro and try to get the Keep symbol right, drop it inside the bubble, back off a good distance and when the Bahro is gone, check to see if he gave you the new Keep symbol. (on the pedestal)

If you have a problem with the keep symbol try to get help from somebody who is good at drawing.

By the way, if you watch the Bahro when he checks your slate and if you see him scratching his head, it means he does not recognize your drawing.