Lie–Leibniz Triple Systems

• Lie–Leibniz triples are a unified ternary framework that integrates Lie triple systems and Leibniz triple systems by relaxing antisymmetry.
• They employ universal envelopes, split and graded decompositions, and cohomological methods to analyze symmetry and derivation identities.
• This framework extends classical Lie theory by facilitating practical insights into ideals, radicals, and higher-arity generalizations in algebra.

Searching arXiv for the cited papers and closely related work on Leibniz triple systems and Lie-type triple structures. arxiv_search query: "Leibniz triple systems split Leibniz triple systems Lie triple systems relative Rota-Baxter (Cao et al., 2014, Bremner et al., 2011)" arxiv_search results requested for:

• "Leibniz triple systems" (Bremner et al., 2011)
• "On split Leibniz triple systems" (Cao et al., 2014)
• "On the structures of split Leibniz triple systems" (Cao et al., 2014)
• "On the structure of graded Leibniz triple systems" (Cao et al., 2016)
• "Lie Admissible Triple Algebras: The Connection Algebra of Symmetric Spaces" (Munthe-Kaas et al., 2023)
• "Generalized metric n-Leibniz algebras and generalized orthogonal representation of metric Lie algebras" (Song et al., 2018)
• "Lie triple system central extensions of Lie algebras" (Kurdiani, 2010)
• "Notes on Cohomologies of Lie Triple Systems" (Zhang, 2014)
• "Cohomology of Leibniz Triple Systems and its applications" (Wu et al., 2021)
• "k-Leibniz algebras from lower order ones: from Lie triple to Lie l-ple systems" (Azcarraga et al., 2013) “Lie–Leibniz triple” (Editor's term) usefully denotes the ternary framework in which Lie triple systems, Leibniz triple systems, and closely related constructions are studied together. In the formal literature, the central object is the Leibniz triple system: a vector space endowed with a trilinear product satisfying the two identities

$$\{a,\{b,c,d\},e\}=\{\{a,b,c\},d,e\}-\{\{a,c,b\},d,e\}-\{\{a,d,b\},c,e\}+\{\{a,d,c\},b,e\},$$

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$

obtained functorially from the identities of Lie triple systems by the Kolesnikov–Pozhidaev algorithm (Bremner et al., 2011). Every Lie triple system is a Leibniz triple system, and every Leibniz algebra yields a Leibniz triple system by the iterated bracket formula $\{x,y,z\}=[[x,y],z]$ (Bremner et al., 2011). This suggests that the Lie–Leibniz triple perspective is best understood not as a single additional axiom system, but as a common ternary domain in which Lie-type symmetry, Leibniz-type derivation identities, root decompositions, cohomology, and higher-arity generalizations are organized.

1. Basic ternary structures

A Lie triple system is a vector space with a trilinear product $[x,y,z]$ satisfying skew-symmetry in the first two variables, the cyclic identity

$[x,y,z]+[y,z,x]+[z,x,y]=0,$

and the derivation identity

$$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$

A Leibniz triple system drops the antisymmetry requirements but retains the Leibniz-type ternary identities displayed above (Bremner et al., 2011).

Two canonical constructions situate Leibniz triple systems between Lie triple systems and Leibniz algebras. First, if $L$ is a Leibniz algebra, then $\{x,y,z\}=[[x,y],z]$ defines a Leibniz triple system structure on $L$. Second, every Lie triple system automatically satisfies the Leibniz triple identities, so Lie triple systems embed as the antisymmetric sector of the Leibniz theory (Cao et al., 2014).

A fundamental internal invariant is the ideal

$$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$

This ideal satisfies

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$0

and $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$1 is a Lie triple system if and only if $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$2 (Cao et al., 2014). Accordingly, $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$3 measures the defect from Lie-type antisymmetry. In the same spirit, a Leibniz triple system is called simple when its product is nonzero and its only ideals are $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$4, $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$5, and $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$6; in the Lie triple case this reduces to the usual notion because $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$7 (Cao et al., 2014).

2. Envelopes, embeddings, and binary realization

The main binary realization of a Leibniz triple system is its universal Leibniz envelope. If $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$8 is a Leibniz triple system, one forms the free Leibniz algebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$9 on the underlying vector space and quotients by the ideal generated by $\{x,y,z\}=[[x,y],z]$0. The resulting Leibniz algebra

$\{x,y,z\}=[[x,y],z]$1

realizes the ternary product as an iterated Leibniz bracket (Bremner et al., 2011). Explicitly, as a vector space,

$\{x,y,z\}=[[x,y],z]$2

with multiplication determined by

$\{x,y,z\}=[[x,y],z]$3

$\{x,y,z\}=[[x,y],z]$4

(Ma et al., 2014). The envelope carries a natural $\{x,y,z\}=[[x,y],z]$5-grading, and the associated involutive automorphism $\{x,y,z\}=[[x,y],z]$6 satisfies

$\{x,y,z\}=[[x,y],z]$7

so $\{x,y,z\}=[[x,y],z]$8 appears as the $\{x,y,z\}=[[x,y],z]$9-eigenspace of $[x,y,z]$0 (Ma et al., 2014).

A different but related construction, crucial for structure theory, is the standard embedding. For a Leibniz triple system $[x,y,z]$1, one considers the two-graded right Leibniz algebra

$[x,y,z]$2

with the bracket on homogeneous elements given by

$[x,y,z]$3

This embedding supplies the ambient Leibniz algebra used to define root spaces, maximal abelian subalgebras, and split decompositions (Cao et al., 2014).

At the level of identities, the enveloping approach is exact: every polynomial identity satisfied by the iterated Leibniz bracket $[x,y,z]$4 in every Leibniz algebra is a consequence of the defining identities of Leibniz triple systems (Bremner et al., 2011). This makes the Leibniz triple system axioms the intrinsic ternary identity theory of iterated Leibniz brackets.

3. Split and graded structure theory

The most developed structural results concern split Leibniz triple systems. Let $[x,y,z]$5 be a maximal abelian subalgebra of the degree-zero part of the standard embedding. For $[x,y,z]$6, the root space of $[x,y,z]$7 is

$[x,y,z]$8

and the system is called split if

$[x,y,z]$9

with $[x,y,z]+[y,z,x]+[z,x,y]=0,$0 and $[x,y,z]+[y,z,x]+[z,x,y]=0,$1 for every $[x,y,z]+[y,z,x]+[z,x,y]=0,$2 (Cao et al., 2014). The root system is symmetric when $[x,y,z]+[y,z,x]+[z,x,y]=0,$3 implies $[x,y,z]+[y,z,x]+[z,x,y]=0,$4.

The triple product is root-additive: whenever $[x,y,z]+[y,z,x]+[z,x,y]=0,$5,

$[x,y,z]+[y,z,x]+[z,x,y]=0,$6

Likewise, root-space brackets in the embedding satisfy the corresponding additive rules (Cao et al., 2014). This permits a connectivity calculus on the root system. Two roots are said to be connected if they can be joined by a finite chain of root sums satisfying the partial-sum conditions of the theory; the resulting equivalence classes define subsystems $[x,y,z]+[y,z,x]+[z,x,y]=0,$7 and, from them, ideals $[x,y,z]+[y,z,x]+[z,x,y]=0,$8 (Cao et al., 2014).

The fundamental decomposition theorem states that for a split Leibniz triple system with symmetric root system,

$[x,y,z]+[y,z,x]+[z,x,y]=0,$9

where $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$0 is a complement of the span of degree-zero triple products and each $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$1 is an ideal attached to a root-connectivity class (Cao et al., 2014). Distinct classes are mutually annihilating:

$$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$2

If the annihilator vanishes and $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$3, the decomposition becomes direct:

$$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$4

(Cao et al., 2014).

An analogous result holds for graded Leibniz triple systems $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$5 over an arbitrary abelian group $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$6. Writing

$$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$7

one defines connectivity on $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$8, obtains equivalence classes $$[x,y,[u,v,w]]=[[x,y,u],v,w]+[u,[x,y,v],w]+[u,v,[x,y,w]].$$9, and constructs ideals $L$0 such that

$L$1

with the same annihilation relations between distinct components (Cao et al., 2016). Under tightness and $L$2, this also becomes a direct sum decomposition.

4. Ideals, simplicity, radicals, and extension theory

Root connectivity constrains ideals very strongly. In the maximal-length setting, where each nonzero root space is one-dimensional, the split theory becomes sharper. A split Leibniz triple system is of maximal length when $L$3 for every $L$4, and it is root-multiplicative when all triple products allowed by the root-addition rules are nonzero in the prescribed sense (Cao et al., 2014). Under the hypotheses

• $L$5,
• $L$6,
• root-multiplicativity,
• $L$7 for all $L$8,
• symmetric $L$9 and $\{x,y,z\}=[[x,y],z]$0,
• and vanishing conditions involving $\{x,y,z\}=[[x,y],z]$1 and $\{x,y,z\}=[[x,y],z]$2,

simplicity is characterized by primeness together with global $\{x,y,z\}=[[x,y],z]$3-connectivity of both $\{x,y,z\}=[[x,y],z]$4 and $\{x,y,z\}=[[x,y],z]$5 (Cao et al., 2014). In less refined form, simplicity forces all nonzero roots to lie in a single connectivity class (Cao et al., 2014).

The intrinsic structure theory also parallels the binary Leibniz theory via the universal envelope. The solvable radical $\{x,y,z\}=[[x,y],z]$6 of a Leibniz triple system is the maximal solvable ideal, and the ternary kernel

$\{x,y,z\}=[[x,y],z]$7

satisfies $\{x,y,z\}=[[x,y],z]$8 and $\{x,y,z\}=[[x,y],z]$9 is a Lie triple system (Ma et al., 2014). The relation with the envelope is precise:

$L$0

where $L$1 is the nilpotent radical (Ma et al., 2014). A Leibniz triple analogue of Levi’s theorem holds:

$L$2

with $L$3 a semisimple subsystem that is in fact a semisimple Lie triple system (Ma et al., 2014).

At the extension level, a perfect Lie algebra $L$4 may be viewed simultaneously as a Lie algebra, a Leibniz algebra, and a Lie triple system via $L$5. Its universal central extension in the category of Lie triple systems is isomorphic to the universal central extension in the Lie category when $L$6, and to the universal central extension in the Leibniz category when $L$7 (Kurdiani, 2010). This places Lie triple extensions directly between the Lie and Leibniz extension theories.

5. Cohomology, deformations, and operator methods

For Lie triple systems, Yamaguti cohomology can be reconstructed from Leibniz cohomology. If $L$8 is a Lie triple system, the fundamental set $L$9 carries a Leibniz algebra structure, and the Yamaguti complex of $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$0 with coefficients in a module $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$1 coincides with the Loday–Pirashvili cohomology complex of the Leibniz algebra $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$2 with coefficients in $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$3 (Zhang, 2014). In this framework, $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$4 classifies abelian extensions, and Nijenhuis operators describe trivial deformations (Zhang, 2014).

For Leibniz triple systems, the relevant low-degree cohomology is concentrated in odd degrees. The first coboundary

$$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$5

shows that 1-cocycles in the adjoint module are derivations (Wu et al., 2021). The third cohomology group $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$6 classifies central extensions of Leibniz triple systems by $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$7, and this classification is implemented explicitly through cocycle-defined extensions on $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$8 (Wu et al., 2021). The same paper develops $$J=\operatorname{span}\{\{a,b,c\}-\{a,c,b\}+\{b,c,a\}:a,b,c\in T\}.$$9-extensions: if $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$00, then

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$01

carries a Leibniz triple system structure, and every even-dimensional quadratic Leibniz triple system is isomorphic to a $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$02-extension under a suitable isotropic-ideal hypothesis (Wu et al., 2021). A quadratic Leibniz triple system $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$03 admits a symplectic form if and only if there exists an invertible derivation $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$04 skew-symmetric with respect to $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$05 (Wu et al., 2021).

The deformation theory of operators on Leibniz triple systems has also been developed. For a representation $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$06 of a Leibniz triple system $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$07 on $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$08, a relative Rota–Baxter operator

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$09

is characterized by

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$10

Such an operator induces a Leibniz triple system structure on $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$11, generates its own cohomology theory, and satisfies the usual deformation pattern: $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$12 controls infinitesimal deformations, while $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$13 carries the obstruction classes for extending order-$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$14 deformations (Wu et al., 2022). The same work shows that cohomology of relative Rota–Baxter operators on Leibniz algebras maps naturally to the associated Leibniz triple system cohomology.

6. Related frameworks and higher generalizations

Several adjacent theories enlarge the Lie–Leibniz triple landscape without collapsing back to the basic Leibniz triple axioms. One direction is the theory of Lie admissible triple algebras. Here an algebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$15 carries a derived triple bracket

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$16

satisfying

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$17

and

$$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$18

The connection algebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$19 of a manifold is such a structure if and only if $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$20 is a locally symmetric space; its triple bracket is then a Lie triple system, and the whole structure embeds canonically into a post-Lie algebra (Munthe-Kaas et al., 2023). This is Lie-type in the ternary bracket and Leibniz-type in the derivation rule for $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$21.

Another direction is metric higher-arity theory. A generalized metric $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$22-Leibniz algebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$23 is in one-to-one correspondence with a Lie triple data $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$24, where $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$25 is a metric Lie algebra, $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$26 is a symmetric non-degenerate $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$27-tensor on $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$28, and $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$29 is a faithful generalized orthogonal representation (Song et al., 2018). This correspondence identifies the $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$30-Leibniz bracket with the action of a Lie algebra of inner derivations and extends to generalized orthogonal derivations and automorphisms (Song et al., 2018).

Higher-order Leibniz constructions also produce generalized Lie-type triple systems. Starting from a metric $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$31-Leibniz algebra, one may construct a $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$32-Leibniz algebra via the Kasymov trace form; imposing appropriate symmetry conditions yields Lie $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$33-ple systems and Lie $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$34-ple systems, both reducing to ordinary Lie triple systems when $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$35 (Azcarraga et al., 2013). At the bialgebraic level, the notion of 3-Leibniz bialgebra extends Leibniz bialgebras to ternary brackets, and there is a theorem relating a 3-Leibniz bialgebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$36 to a binary Leibniz bialgebra $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$37 on its associated Leibniz algebra (Rezaei-Aghdam et al., 2016).

Taken together, these developments support a precise reading of the Lie–Leibniz triple theme. Lie triple systems provide the antisymmetric benchmark; Leibniz triple systems provide the fundamental non-antisymmetric ternary generalization; split, graded, cohomological, metric, and higher-arity theories show that much of Lie-theoretic structure survives after antisymmetry is relaxed, but typically survives in modified form through the ideals $$\{a,b,\{c,d,e\}\}=\{\{a,b,c\},d,e\}-\{\{a,b,d\},c,e\}-\{\{a,b,e\},c,d\}+\{\{a,b,e\},d,c\},$$38, root connectivity, enveloping Leibniz algebras, and duality with associated binary Leibniz structures (Cao et al., 2014).