$p$-divisible group of tori

Question

I am looking for a reference of the following question which should be well known. Let $k$ be any field and $T$ an algebraic torus over $k$ which is not necessarily split. Let $T(l)$ be the $l$-divisible group associated with $T$ with $l$ a prime number. Is it true that the height of $T(l)$ is equal to the rank of the torus $T$? I don't know if $l$ should be coprime with the characteristic of $k$. In the split case this is true without restriction on $l$ but I don't know if descent works.