Don't lead your new tamed eel home while skimming the surface on your "whale"... The eel will surface itself at some point and instamtly die from being out of the water. Took me 3 eels this way before I figured out how they were dying without getting into combat.
Don't ever tame one of those, unless you're trying to get the achievment. Wild ones are very annoying and that makes you think tamed ones will be useful, right? NO, THEY'RE NOT! They do have an aoe attack that stuns, but they're not rideable and the AI won't ever use this attack. Instead, they will keep bitting with a ridiculous damage, like 6 or 7 points.
One of the most feared aquatic wild creatures is totally useless when tamed.