bilinear operation

In each case the significance of the symbol is the same: the freest bilinear operation.

Algebra", when referring to a structure, often means a vector space or module equipped with an additional bilinear operation."

The product rule of differential calculus applies to any bilinear operation, and therefore also to the cross product:

In mathematics, the seven-dimensional cross product is a bilinear operation on vectors in seven-dimensional Euclidean space.

A set of all nonclosed 1-forms together with the bilinear operation is a Valya algebra, and it is not a Lie algebra.

Furthermore, if R is a commutative superalgebra, then supermatrix multiplication is a bilinear operation, so that M(R) forms a superalgebra over R.

A Lie algebra over a field F is a vector space over F equipped with a skew-symmetric bilinear operation called the Lie bracket, which satisfies the Jacobi identity.

The Poisson bracket on (M, ω) is a bilinear operation on differentiable functions, defined by ; the Poisson bracket of two functions on M is itself a function on M.

This bilinear operation is actually the zero map, but the second derivative, under the proper identification of tangent spaces, yields an operation that satisfies the axioms of a Lie bracket, and it is equal to twice the one defined through left-invariant vector fields.