If $L$ is a semi-simple Lie algebra, then $L=L'$.   Since $L$ is semi-simple we can write it as a direct sum of simple ideals $L_i$, i.e. $L=\oplus_{i=1}^r L_i$. Then $L'=\oplus_{i=1}^r L_i'$ and since every $L_i$ is simple we have $L_i'=L_i$ and consequently $$L'=\oplus_{i=1}^r L_i'=\oplus_{i=1}^r L_i=L$$  This proof seems surprisingly short. So my question is, did I make any mistakes or did I forget any crucial assumptions? Is there perhaps a more elementary proof?
Can anyone explain why a semi-simple finite dimensional Lie algebra $\mathfrak{g}$ has to be perfect ?   The natural way to prove something like that would be to look to the algebra generated by the Lie brackets, which when $\mathfrak{g}$ is not perfect would be expected to be solvable. But it doesn't seem to work.
Why is my StackOverflow with an indent and doesn't have reputation TeX editor with line-spacing preferences Cannot Buy Land in Falkreath Fastest way to clamp a real (fixed/floating point) value? Is there a more efficient way to clamp real numbers than using if statements or ternary operators? I want to do this both for doubles and for a 32-bit fixpoint implementation (16.16). I'm not asking for code that can handle both cases; they will be handled in separate functions.  Obviously, I can do something like:  double clampedA; double a = calculate(); clampedA = a &gt; MY_MAX ? MY_MAX : a; clampedA = a &lt; MY_MIN ? MY_MIN : a;   or  double a = calculate(); double clampedA = a; if(clampedA &gt; MY_MAX)     clampedA = MY_MAX; else if(clampedA &lt; MY_MIN)     clampedA = MY_MIN;   The fixpoint version would use functions/macros for comparisons.  This is done in a performance-critical part of the code, so I'm looking for an as efficient way to do it as possible (which I suspect would involve bit-manipulation)   EDIT: It has to be standard/portable C, platform-specific functionality is not of any interest here. Also, MY_MIN and MY_MAX are the same type as the value I want clamped (doubles in the examples above). What do "." and ".." mean when in a folder? when I used ls -a to view hidden files in folder there were 2 hidden files created by default what is the significance of this two files . and ..?  ls -a   Produces  .  .. Stop setInterval call in JavaScript What is a good reference for rigorous Electromagnetism and Electrodynamics? Is there any good book on Electromagnetism from a more mathematical point of view? By this I mean a book which makes use of differential forms and maybe De Rham cohomology. I was also searching for something that would include E&amp;M in the big picture of Gauge theories. 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