  Let $X$ be an $n$-dimensional smooth Fano complex variety of Picard number
one. Assume that the VMRT at a general point of $X$ is smooth irreducible and
non-degenerate (which holds if $X$ is covered by lines with index $ >(n+2)/2$).
It is proven that $\dim \mathfrak{aut}(X) > n(n+1)/2$ if and only if $X$ is
isomorphic to $\mathbb{P}^n, \mathbb{Q}^n$ or ${\rm Gr}(2,5)$. Furthermore, the
equality $\dim \mathfrak{aut}(X) = n(n+1)/2$ holds only when $X$ is isomorphic
to the 6-dimensional Lagrangian Grassmannian ${\rm Lag}(6)$ or a general
hyperplane section of ${\rm Gr}(2,5)$.
