Why do Gell-Mann matrices have this normalization?

This might be a stupid question, but why is the normalization of the Gell-Mann matrices (basis of the $mathrm{su}(3)$ Lie algebra) chosen to be
$$mathrm{trace}(lambda_ilambda_j)=2delta_{ij}$$
instead of just $delta_{ij}$ without the factor $2$? In most of linear-algebra, basis vectors are normalized to $1$ (or not normalized at all). Why not in the context of Lie Algebras? Is there a way of looking at this which makes the factor $2$ seem natural?

On a related note, some physics texts change the normalization by defining "the generators of the $mathrm{SU}(3)$ group" as $T_i=frac{1}{2}lambda_i$. But these just fulfil $mathrm{trace}(T_iT_j)=frac{1}{2}delta_{ij}$ which seems just as unnatural to me. (And the difference between these two normalization conventions just cost me an hour of chasing a missing factor $4$ in a long calculation. Which is why I’m asking this question xD).