how can i see that $j\beta =\beta j$ is equivalent to $\beta^*+\beta$ being a multiple of the identity?

Question

A vector field $X$ is said to be conformal if $L_Xj=0$ where j is the almost complex structure. The conformality condition is equivalent to $j\beta =\beta j$. Where $\beta : ker(\theta) \mapsto ker(\theta)$ such that $\beta(u)= \nabla_uX$ and $\theta$ is contact form such that $\theta(X)=1$. \ my question is how can i see that $j\beta =\beta j$ is equivalent to $\beta^*+\beta$ being a multiple of the identity? \ What are the conditions so that i can see $ j=\beta-\beta^*$?