Irreducible orthogonal decompositions in Lie algebras

The weakened Winnie-the-Pooh problem on irreducible orthogonal decompositions (IOD's) of a simple finite-dimensional complex Lie algebra $\mathscr L$ (i.e., orthogonal decompositions of $\mathscr L$ whose automorphism group acts on $\mathscr L$ absolutely irreducibly is solved). It is proved that Lie algebras of types $A_{p-2}$ ($p$ a prime number, $p\ne2^d+1$), $C_3$ and $E_7$ have no IOD's. All IOD's of Lie algebras of types $A_{p-1}$ ($p$ is a prime number), $G_2$, $F_4$, $E_6$ and $E_8$ are found.
Bibliography: 25 titles.