Let $Q$ be the smooth quadric threefold in $\mathbb{P}^4_{\mathbb{C}}$ defined by the equation $x_0x_4+x_1x_3+x_2^2=0$.

Is it true that the automorphism group of $Q$ is $SO(Q;\mathbb{C})$ which is isomorphic to $SO(5;\mathbb{C})$?

How can I prove $Aut(Q)=SO(Q)$? And can you give a concrete isomorphism between $SO(Q)$ and $SO(5)$?