A Lyapunov-like (linear) transformation L on a Euclidean Jordan algebra V is defined by the condition $$x \in K,\enspace y\in K^*,\quad \langle x,y\rangle =0\Rightarrow \langle L(x),y\rangle=0,$$where K is the symmetric cone of V. In this paper, we give an elementary proof (avoiding Lie...