On the existence of orthogonal decompositions of the simple Lie algebra of type C3

Orthogonal decompositions (OD:s) that are monomial have been constructed for most simple Lie algebras but in some cases the existence of an OD is still an open question. One of them is Lie algebras of type Cn where n is not a power of 2. In this paper we use computational methods to prove that C3 has no monomial OD. 1 Orthogonal decompositions of simple Lie algebras The basic question, of which we will study a special case, is whether all classical simple Lie algebras can be decomposed into a direct sum of Cartan subalgebras which are orthogonal to each other in the sense that the Killing form B(X, Y ) = 0 if X and Y are elements from different Cartan subalgebras occurring in the decomposition. Such a direct sum is called an orthogonal decomposition (OD). This problem has been studied for all simple Lie algebras over C, and OD:s have been constructed in all cases except for An when n + 1 is not a prime power and Cn when n is not a power of 2 [3]. In the latter cases the existence of an OD is an open question. One result pointing to a negative answer is that A5 has no OD of monomial type [4]. The main result of this paper is that C3 has no monomial orthogonal decomposition either.