  The paper studies the global convergence of the Jacobi method for symmetric
matrices of size $4$. We prove global convergence for all $720$ cyclic pivot
strategies. Precisely, we show that inequality $S(A^{[t+3]})\leq\gamma
S(A^{[t]})$, $t\geq1$, holds with the constant $\gamma<1$ that depends neither
on the matrix $A$ nor on the pivot strategy. Here $A^{[t]}$ stands for the
matrix obtained from $A$ after $t$ full cycles of the Jacobi method and $S(A)$
is the off-diagonal norm of $A$. We show why three consecutive cycles have to
be considered. The result has a direct application on the $J$-Jacobi method.
