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There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more preety or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On $q$-deformed infinite-dimensional $n$-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more pretty or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On $q$-deformed infinite-dimensional $n$-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more preety or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On $q$-deformed infinite-dimensional $n$-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more preety or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On $q$-deformed infinite-dimensional $n$-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more preety or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On q-deformed infinite-dimensional n-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]

There are various deformations of the Jacobi identity that can be found scattered in the literature. As far as i know, using the definition: $[A,B]_q=AB-qBA$, one of the most general ones (though i do not now if this is "symmetric" enough for your purposes) is the following one: $$ \big[A,[B,C]_{q_1}\big]_{q_2}+q_2\big[B,[C,A]_{q_1}\big]_{q_2^{-1}}+\big[C,[A,B]_{q_1q_2}\big]=0 $$ which is valid for arbitrary values of the parameters $q_1$, $q_2$. See [1].

Notice that both identities stated by the OP are compatible with it:

• The first one can be recovered for: $q_1=q$, $q_2=1$,
• while a particular case (for $k=-m$) of the second one for: $q_1=q^n$, $q_2=q^m$.

Edit: I was thinking that the second identity of the OP might be combined with the 2-parameter deformation given here to construct a further 3-parameter deformation of the Jacobi identity. But i am not sure on its exact form, i 'll try to think a little on it and come back if something interesting comes up.

Edit 2: I was thinking about the OP's last paragraph (asking for something more preety or symmetric). Well, i did some further search and it turns out that we can get some symmetric generalization if we consider a two parameter deformation of the commutator. Using $[A,B]_{(p,q)}=pAB-qBA$ we have the following 3-parameter deformation of the Jacobi identity: $$ {\small \big[A,[B,C]_{(q_1,q_1^{-1})}\big]_{(q_3/q_2,q_2/q_3)}+\big[B,[C,A]_{(q_2,q_2^{-1})}\big]_{(q_1/q_3,q_3/q_1)}+\big[C,[A,B]_{(q_3,q_3^{-1})}\big]_{(q_2/q_1,q_1/q_2)}=0 } $$ See: [2].

References:
[1]. $q$-deformed Jacobi identity, $q$-oscillators and $q$-deformed infinite-dimensional algebras, Chaichian, Kulish, Lukierski, Phys. Let. B, v.237, 3-4, p. 401-406, 1990
[2]. On $q$-deformed infinite-dimensional $n$-algebra Lu Ding, Xiao-Yu Jia, Ke Wu, Zhao-Wen Yan, Wei-Zhong Zhao, arXiv:1404.0464v3 [hep-th]