Definition 7.3 :
Let
M
{\displaystyle M}
C
n
{\displaystyle {\mathcal {C}}^{n}}
vector field Lie bracket , denoted by
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
[
⋅
,
⋅
]
:
X
(
M
)
×
X
(
M
)
→
X
(
M
)
,
[
V
,
W
]
(
p
)
(
φ
)
:=
V
(
p
)
(
W
φ
)
−
W
(
p
)
(
V
φ
)
{\displaystyle [\cdot ,\cdot ]:{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M),[\mathbf {V} ,\mathbf {W} ](p)(\varphi ):=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )}
Theorem 6.4 : If
V
,
W
{\displaystyle \mathbf {V} ,\mathbf {W} }
C
n
{\displaystyle {\mathcal {C}}^{n}}
M
{\displaystyle M}
[
V
,
W
]
{\displaystyle [\mathbf {V} ,\mathbf {W} ]}
C
n
{\displaystyle {\mathcal {C}}^{n}}
M
{\displaystyle M}
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
X
(
M
)
{\displaystyle {\mathfrak {X}}(M)}
Proof :
1. We show that for each
p
∈
M
{\displaystyle p\in M}
[
V
,
W
]
(
p
)
∈
T
(
p
)
M
{\displaystyle [\mathbf {V} ,\mathbf {W} ](p)\in T(p)M}
φ
,
ϑ
∈
C
∞
(
M
)
{\displaystyle \varphi ,\vartheta \in {\mathcal {C}}^{\infty }(M)}
c
∈
R
{\displaystyle c\in \mathbb {R} }
1.1 We prove linearity:
[
V
,
W
]
(
p
)
(
φ
+
c
ϑ
)
=
V
(
p
)
(
W
(
φ
+
c
ϑ
)
)
−
W
(
p
)
(
V
(
φ
+
c
ϑ
)
)
=
V
(
p
)
(
W
φ
+
c
W
ϑ
)
−
W
(
p
)
(
V
φ
+
c
V
ϑ
)
=
V
(
p
)
(
W
φ
)
−
W
(
p
)
(
V
φ
)
+
c
(
V
(
p
)
(
W
ϑ
)
−
W
(
p
)
(
V
ϑ
)
)
=
[
V
,
W
]
(
p
)
(
φ
)
+
c
[
V
,
W
]
(
p
)
(
ϑ
)
{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi +c\vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi +c\vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi +c\vartheta ))\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \varphi +c\mathbf {V} \vartheta )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c(\mathbf {V} (p)(\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \vartheta ))\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )\end{aligned}}}
1.2 We prove the product rule:
[
V
,
W
]
(
p
)
(
φ
ϑ
)
=
V
(
p
)
(
W
(
φ
ϑ
)
)
−
W
(
p
)
(
V
(
φ
ϑ
)
)
=
V
(
p
)
(
φ
W
ϑ
+
ϑ
W
φ
)
−
W
(
p
)
(
φ
V
ϑ
+
ϑ
V
φ
)
=
V
(
p
)
(
φ
W
ϑ
)
+
V
(
p
)
(
ϑ
W
φ
)
−
W
(
p
)
(
φ
V
ϑ
)
−
W
(
p
)
(
ϑ
V
φ
)
=
φ
(
p
)
V
(
p
)
(
W
ϑ
)
+
(
Y
ϑ
)
(
p
)
⏞
=
Y
(
p
)
(
ϑ
)
V
(
p
)
(
φ
)
+
ϑ
(
p
)
V
(
p
)
(
W
φ
)
+
(
Y
φ
)
(
p
)
⏞
=
Y
(
p
)
(
φ
)
V
(
p
)
(
ϑ
)
−
φ
(
p
)
W
(
p
)
(
V
ϑ
)
−
(
X
ϑ
)
(
p
)
⏞
=
X
(
p
)
(
ϑ
)
W
(
p
)
(
φ
)
−
ϑ
(
p
)
W
(
p
)
(
V
φ
)
−
(
X
φ
)
(
p
)
⏞
=
X
(
p
)
(
φ
)
W
(
p
)
(
ϑ
)
=
φ
(
p
)
V
(
p
)
(
W
ϑ
)
−
φ
(
p
)
W
(
p
)
(
V
ϑ
)
+
ϑ
(
p
)
V
(
p
)
(
W
φ
)
−
ϑ
(
p
)
W
(
p
)
(
V
φ
)
=
φ
(
p
)
[
V
,
W
]
(
p
)
(
ϑ
)
+
ϑ
(
p
)
[
V
,
W
]
(
p
)
(
φ
)
{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi \vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi \vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi \vartheta ))\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta +\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta +\vartheta \mathbf {V} \varphi )\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta )+\mathbf {V} (p)(\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta )-\mathbf {W} (p)(\vartheta \mathbf {V} \varphi )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )+\overbrace {\mathbf {(} Y\vartheta )(p)} ^{=Y(p)(\vartheta )}\mathbf {V} (p)(\varphi )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )+\overbrace {\mathbf {(} Y\varphi )(p)} ^{=Y(p)(\varphi )}\mathbf {V} (p)(\vartheta )\\&~~~~-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )-\overbrace {\mathbf {(} X\vartheta )(p)} ^{=X(p)(\vartheta )}\mathbf {W} (p)(\varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )-\overbrace {\mathbf {(} X\varphi )(p)} ^{=X(p)(\varphi )}\mathbf {W} (p)(\vartheta )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )\\&=\varphi (p)[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )+\vartheta (p)[\mathbf {V} ,\mathbf {W} ](p)(\varphi )\end{aligned}}}
2. We show that
[
V
,
W
]
{\displaystyle [\mathbf {V} ,\mathbf {W} ]}
C
n
{\displaystyle {\mathcal {C}}^{n}}
Let
φ
∈
C
n
(
M
)
{\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
V
,
W
{\displaystyle \mathbf {V} ,\mathbf {W} }
C
n
{\displaystyle {\mathcal {C}}^{n}}
V
φ
{\displaystyle \mathbf {V} \varphi }
W
φ
{\displaystyle \mathbf {W} \varphi }
C
n
(
M
)
{\displaystyle {\mathcal {C}}^{n}(M)}
V
,
W
{\displaystyle \mathbf {V} ,\mathbf {W} }
C
n
{\displaystyle {\mathcal {C}}^{n}}
V
(
W
φ
)
{\displaystyle \mathbf {V} (\mathbf {W} \varphi )}
W
(
V
φ
)
{\displaystyle \mathbf {W} (\mathbf {V} \varphi )}
C
n
(
M
)
{\displaystyle {\mathcal {C}}^{n}(M)}
[
V
,
W
]
φ
{\displaystyle [\mathbf {V} ,\mathbf {W} ]\varphi }
C
n
(
M
)
{\displaystyle {\mathcal {C}}^{n}(M)}
φ
{\displaystyle \varphi }
[
V
,
W
]
{\displaystyle [\mathbf {V} ,\mathbf {W} ]}
C
n
{\displaystyle {\mathcal {C}}^{n}}
◻
{\displaystyle \Box }
Theorem 6.5 :
If
M
{\displaystyle M}
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
X
(
M
)
{\displaystyle {\mathfrak {X}}(M)}
[
⋅
,
⋅
]
{\displaystyle [\cdot ,\cdot ]}
Proof :
1. First we note that
X
(
M
)
{\displaystyle {\mathfrak {X}}(M)}
2. Second, we prove that for the vector Lie bracket, the three calculation rules of definition 6.1 are satisfied. Let
V
,
W
,
U
∈
X
(
M
)
{\displaystyle \mathbf {V} ,\mathbf {W} ,\mathbf {U} \in {\mathfrak {X}}(M)}
c
∈
R
{\displaystyle c\in \mathbb {R} }
2.1 We prove bilinearity. For all
p
∈
M
{\displaystyle p\in M}
φ
∈
C
n
(
M
)
{\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
[
V
,
W
+
c
U
]
(
p
)
(
φ
)
=
V
(
p
)
(
(
W
+
c
U
)
φ
)
−
(
W
+
c
U
)
(
p
)
(
V
φ
)
=
V
(
p
)
(
W
φ
+
c
U
φ
)
−
W
(
p
)
(
V
φ
)
−
c
U
(
p
)
(
V
φ
)
=
V
(
p
)
(
W
φ
)
−
W
(
p
)
(
V
φ
)
+
c
V
(
p
)
(
U
φ
)
−
c
U
(
p
)
(
V
φ
)
=
[
V
,
W
]
(
p
)
(
φ
)
+
c
[
V
,
U
]
(
p
)
(
φ
)
{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]}(p)(\varphi )&=\mathbf {V} (p)((\mathbf {W} +c\mathbf {U} )\varphi )-(\mathbf {W} +c\mathbf {U} )(p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {U} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c\mathbf {V} (p)(\mathbf {U} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {U} ](p)(\varphi )\end{aligned}}}
and hence, since
p
∈
M
{\displaystyle p\in M}
φ
∈
C
n
(
M
)
{\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
[
V
,
W
+
c
U
]
=
[
V
,
W
]
+
c
[
V
,
U
]
{\displaystyle [\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]=[\mathbf {V} ,\mathbf {W} ]+c[\mathbf {V} ,\mathbf {U} ]}
Analogously (see exercise 1), it can be proven that
[
V
+
c
W
,
U
]
=
[
V
,
U
]
+
c
[
W
,
U
]
{\displaystyle [\mathbf {V} +c\mathbf {W} ,\mathbf {U} ]=[\mathbf {V} ,\mathbf {U} ]+c[\mathbf {W} ,\mathbf {U} ]}
2.2 We prove skew-symmetry. We have for all
p
∈
M
{\displaystyle p\in M}
φ
∈
C
n
(
M
)
{\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
[
V
,
W
]
(
p
)
(
φ
)
=
V
(
p
)
(
W
φ
)
−
W
(
p
)
(
V
φ
)
=
−
(
W
(
p
)
(
V
φ
)
−
V
(
p
)
(
W
φ
)
)
=
−
[
W
,
V
]
(
p
)
(
φ
)
{\displaystyle [\mathbf {V} ,\mathbf {W} ](p)(\varphi )=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )=-(\mathbf {W} (p)(\mathbf {V} \varphi )-\mathbf {V} (p)(\mathbf {W} \varphi ))=-[\mathbf {W} ,\mathbf {V} ](p)(\varphi )}
2.3 We prove Jacobi's identity. We have for all
p
∈
M
{\displaystyle p\in M}
φ
∈
C
n
(
M
)
{\displaystyle \varphi \in {\mathcal {C}}^{n}(M)}
[
V
,
[
W
,
U
]
]
(
p
)
(
φ
)
+
[
U
,
[
V
,
W
]
]
(
p
)
(
φ
)
+
[
W
,
[
U
,
V
]
]
(
p
)
(
φ
)
=
V
(
p
)
(
[
W
,
U
]
φ
)
−
[
W
,
U
]
(
p
)
(
V
φ
)
+
U
(
p
)
(
[
V
,
W
]
φ
)
−
[
V
,
W
]
(
p
)
(
U
φ
)
+
W
(
p
)
(
[
U
,
V
]
φ
)
−
[
U
,
V
]
(
p
)
(
W
φ
)
=
V
(
p
)
(
W
(
U
φ
)
−
U
(
W
φ
)
)
−
W
(
p
)
(
U
(
V
φ
)
)
+
U
(
p
)
(
W
(
V
φ
)
)
+
U
(
p
)
(
V
(
W
φ
)
−
W
(
V
φ
)
)
−
V
(
p
)
(
W
(
U
φ
)
)
+
W
(
p
)
(
V
(
U
φ
)
)
+
W
(
p
)
(
U
(
V
φ
)
−
V
(
U
φ
)
)
−
U
(
p
)
(
V
(
W
φ
)
)
+
V
(
p
)
(
U
(
W
φ
)
)
=
0
{\displaystyle {\begin{aligned}{[\mathbf {V} ,[\mathbf {W} ,\mathbf {U} ]]}(p)(\varphi )+[\mathbf {U} ,[\mathbf {V} ,\mathbf {W} ]](p)(\varphi )+[\mathbf {W} ,[\mathbf {U} ,\mathbf {V} ]](p)(\varphi )&=\mathbf {V} (p)([\mathbf {W} ,\mathbf {U} ]\varphi )-[\mathbf {W} ,\mathbf {U} ](p)(\mathbf {V} \varphi )\\&~~~~+\mathbf {U} (p)([\mathbf {V} ,\mathbf {W} ]\varphi )-[\mathbf {V} ,\mathbf {W} ](p)(\mathbf {U} \varphi )\\&~~~~+\mathbf {W} (p)([\mathbf {U} ,\mathbf {V} ]\varphi )-[\mathbf {U} ,\mathbf {V} ](p)(\mathbf {W} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi )-\mathbf {U} (\mathbf {W} \varphi ))-\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi ))+\mathbf {U} (p)(\mathbf {W} (\mathbf {V} \varphi ))\\&~~~~+\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi )-\mathbf {W} (\mathbf {V} \varphi ))-\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi ))+\mathbf {W} (p)(\mathbf {V} (\mathbf {U} \varphi ))\\&~~~~+\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi )-\mathbf {V} (\mathbf {U} \varphi ))-\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi ))+\mathbf {V} (p)(\mathbf {U} (\mathbf {W} \varphi ))\\&=0\end{aligned}}}
, where the last equality follows from the linearity of
V
(
p
)
,
W
(
p
)
{\displaystyle \mathbf {V} (p),\mathbf {W} (p)}
U
(
p
)
{\displaystyle \mathbf {U} (p)}
◻
{\displaystyle \Box }
![{\displaystyle [\cdot ,\cdot ]:L\times L\to L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65ed0dcd16a5bcb57a3b06e7321fb2a9b5ddcaed)
![{\displaystyle [\mathbf {u} ,\mathbf {v} +b\mathbf {w} ]=[\mathbf {u} ,\mathbf {v} ]+b[\mathbf {u} ,\mathbf {w} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b328f3552e4ee8df7341d3b5616eb69cb2425e9c)
![{\displaystyle [\mathbf {u} +b\mathbf {v} ,\mathbf {w} ]=[\mathbf {u} ,\mathbf {w} ]+b[\mathbf {v} ,\mathbf {w} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f7a0e7dcb4c8aa4e15e0214321d97c80af93d32e)
![{\displaystyle [\mathbf {u} ,\mathbf {v} ]=-[\mathbf {v} ,\mathbf {u} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01271176aef6a1bb1827f7f2a6256e6e2835a076)
![{\displaystyle [\mathbf {u} ,[\mathbf {v} ,\mathbf {w} ]]+[\mathbf {w} ,[\mathbf {u} ,\mathbf {v} ]]+[\mathbf {v} ,[\mathbf {w} ,\mathbf {u} ]]=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aae7f256756d7040c04b23c728a96e3dd1343d49)
![{\displaystyle [\cdot ,\cdot ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/28dd4c22d60192519c1c12cf645b040f368db9e9)
![{\displaystyle [\cdot ,\cdot ]:{\mathfrak {X}}(M)\times {\mathfrak {X}}(M)\to {\mathfrak {X}}(M),[\mathbf {V} ,\mathbf {W} ](p)(\varphi ):=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1c0a1cad4e8d7ac44d9f7d6e81059e2de21f21af)
![{\displaystyle [\mathbf {V} ,\mathbf {W} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3eb486b8d35d668027c3a45a2477bed67b6a42dd)
\in T(p)M}](https://wikimedia.org/api/rest_v1/media/math/render/svg/35dae1b5d03f0bf31294f22933ee071a526b5790)
![{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi +c\vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi +c\vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi +c\vartheta ))\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \varphi +c\mathbf {V} \vartheta )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c(\mathbf {V} (p)(\mathbf {W} \vartheta )-\mathbf {W} (p)(\mathbf {V} \vartheta ))\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646d4d61a3421240fb6b14546af7f0f538d7fbdc)
![{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} ]}(p)(\varphi \vartheta )&=\mathbf {V} (p)(\mathbf {W} (\varphi \vartheta ))-\mathbf {W} (p)(\mathbf {V} (\varphi \vartheta ))\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta +\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta +\vartheta \mathbf {V} \varphi )\\&=\mathbf {V} (p)(\varphi \mathbf {W} \vartheta )+\mathbf {V} (p)(\vartheta \mathbf {W} \varphi )-\mathbf {W} (p)(\varphi \mathbf {V} \vartheta )-\mathbf {W} (p)(\vartheta \mathbf {V} \varphi )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )+\overbrace {\mathbf {(} Y\vartheta )(p)} ^{=Y(p)(\vartheta )}\mathbf {V} (p)(\varphi )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )+\overbrace {\mathbf {(} Y\varphi )(p)} ^{=Y(p)(\varphi )}\mathbf {V} (p)(\vartheta )\\&~~~~-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )-\overbrace {\mathbf {(} X\vartheta )(p)} ^{=X(p)(\vartheta )}\mathbf {W} (p)(\varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )-\overbrace {\mathbf {(} X\varphi )(p)} ^{=X(p)(\varphi )}\mathbf {W} (p)(\vartheta )\\&=\varphi (p)\mathbf {V} (p)(\mathbf {W} \vartheta )-\varphi (p)\mathbf {W} (p)(\mathbf {V} \vartheta )+\vartheta (p)\mathbf {V} (p)(\mathbf {W} \varphi )-\vartheta (p)\mathbf {W} (p)(\mathbf {V} \varphi )\\&=\varphi (p)[\mathbf {V} ,\mathbf {W} ](p)(\vartheta )+\vartheta (p)[\mathbf {V} ,\mathbf {W} ](p)(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1cf5af860ab89b081ad4ad4e950cf527db80b6e9)
![{\displaystyle [\mathbf {V} ,\mathbf {W} ]\varphi }](https://wikimedia.org/api/rest_v1/media/math/render/svg/bc53de15db4be5b981591c391004bc2e8d35c692)
![{\displaystyle {\begin{aligned}{[\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]}(p)(\varphi )&=\mathbf {V} (p)((\mathbf {W} +c\mathbf {U} )\varphi )-(\mathbf {W} +c\mathbf {U} )(p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi +c\mathbf {U} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )+c\mathbf {V} (p)(\mathbf {U} \varphi )-c\mathbf {U} (p)(\mathbf {V} \varphi )\\&=[\mathbf {V} ,\mathbf {W} ](p)(\varphi )+c[\mathbf {V} ,\mathbf {U} ](p)(\varphi )\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/42a62381b287abbc3f647290a9bca8a4cfd1e8b0)
![{\displaystyle [\mathbf {V} ,\mathbf {W} +c\mathbf {U} ]=[\mathbf {V} ,\mathbf {W} ]+c[\mathbf {V} ,\mathbf {U} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a6d251cc931149bdd8a168a939a174858cbe2f09)
![{\displaystyle [\mathbf {V} +c\mathbf {W} ,\mathbf {U} ]=[\mathbf {V} ,\mathbf {U} ]+c[\mathbf {W} ,\mathbf {U} ]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2daafcb0712d4ec407948a6e69d109bb120af9e2)
(\varphi )=\mathbf {V} (p)(\mathbf {W} \varphi )-\mathbf {W} (p)(\mathbf {V} \varphi )=-(\mathbf {W} (p)(\mathbf {V} \varphi )-\mathbf {V} (p)(\mathbf {W} \varphi ))=-[\mathbf {W} ,\mathbf {V} ](p)(\varphi )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d8a8dde36f29d7a13feccdb82ab75697437f5ff)
![{\displaystyle {\begin{aligned}{[\mathbf {V} ,[\mathbf {W} ,\mathbf {U} ]]}(p)(\varphi )+[\mathbf {U} ,[\mathbf {V} ,\mathbf {W} ]](p)(\varphi )+[\mathbf {W} ,[\mathbf {U} ,\mathbf {V} ]](p)(\varphi )&=\mathbf {V} (p)([\mathbf {W} ,\mathbf {U} ]\varphi )-[\mathbf {W} ,\mathbf {U} ](p)(\mathbf {V} \varphi )\\&~~~~+\mathbf {U} (p)([\mathbf {V} ,\mathbf {W} ]\varphi )-[\mathbf {V} ,\mathbf {W} ](p)(\mathbf {U} \varphi )\\&~~~~+\mathbf {W} (p)([\mathbf {U} ,\mathbf {V} ]\varphi )-[\mathbf {U} ,\mathbf {V} ](p)(\mathbf {W} \varphi )\\&=\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi )-\mathbf {U} (\mathbf {W} \varphi ))-\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi ))+\mathbf {U} (p)(\mathbf {W} (\mathbf {V} \varphi ))\\&~~~~+\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi )-\mathbf {W} (\mathbf {V} \varphi ))-\mathbf {V} (p)(\mathbf {W} (\mathbf {U} \varphi ))+\mathbf {W} (p)(\mathbf {V} (\mathbf {U} \varphi ))\\&~~~~+\mathbf {W} (p)(\mathbf {U} (\mathbf {V} \varphi )-\mathbf {V} (\mathbf {U} \varphi ))-\mathbf {U} (p)(\mathbf {V} (\mathbf {W} \varphi ))+\mathbf {V} (p)(\mathbf {U} (\mathbf {W} \varphi ))\\&=0\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f1aa3daa878ce7bb96e90f0f6a839de2f11f94)
