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Take $d$ a strictly positive integer, and consider the (proper) Euclidean group $E^d$ (the symmetry group of $\mathbf{R}^d$ with the conventional inner product), and the (proper, ortochronous) Poincaré group $P^d$ (symmetry group of the Minkowski space $M^d$).

I need a reference for the following fact.

Take an unit vector $e \in \mathbf{R}^d$ and let $E^{d-1}_e$ be the Euclidean subgroup which fixes that direction. Then the Lie algebra $\mathfrak{e}^d$ of $E^d$ can be decomposed as $\mathfrak{e}^d = \mathfrak{e}^{d-1}_e \oplus \mathfrak{m}^{d-1}_e$ where $\mathfrak{e}^{d-1}_e$ is the Lie algebra of $E^{d-1}_e$ and $\mathfrak{m}^{d-1}_e$ is another appropriate Lie algebra.

Also, the Poincaré algebra can be formally written as $\mathfrak{p}^d = \mathfrak{e}^{d-1}_e \oplus i\mathfrak{m}^{d-1}_e$.

It looks like it has something to do with those Wick rotations, and most of the references I found only treat the Poincaré algebra... Some help would be appreciated!

Thank you.

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Suppose that you drop one dimension from both groups: time in the Poincaré case, one spatial dimension in the Euclidean (since all directions are the same in that case).

Then you obtain two $(d-1)$-dimensional spaces that have exactly the same metric and, therefore, the same transformation groups.

(By the way, this is the reason why we are so familiar with 3-dimensional rotations.)

If you add back the last dimension, in both cases, what happens?

In the Euclidean case, the added dimension is linked to the previous ones by a rotation. In the Lorentz case, by a boost (since we have said we added time).

As you probably know, at the Lie algebra level the difference between a boost and a rotation is just a $i$ factor. (Think of the difference between $\sin, \cos$ and $\sinh, \cosh$.)

Therefore, whatever was added in the Euclidean case is the same in the Poincaré case, but with a factor of $i$.

(Translations here don't matter at all.)

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