Laplace Operator and Polynomial Invariants

LetAbe a finite dimensional simple algebra (not necessarily associative) over the field of complex numbersC, and letGdenote the automorphism group Aut(A). Suppose thatAhas a symmetric nondegenerate associativeG-invariant bilinear form x, y and a compact real form, i.e., a subalgebraBoverRof dimension dimRB = dimCA, whereAis equal to the span ofBoverCand the restriction of x, y toBis positive definite. We describe generators of the algebra of polynomialG-invariants of a system of several vectors fromAin terms of x, y and Laplace operators. In particular, we give generators of the algebra of polynomial invariants of the adjoint representation of a simple linear algebraic group of any exceptional type ≠ E6. As a consequence, we get the First Main Theorem on matrix invariants, invariants of minimal representation ofG2andF4.